group invariance in statistical inference (narayan)
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Group Invariance in
Statistical Inference
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G r o u p I n v a r i a n c e i n
Stat is t ical Inference
N a r a y a n C . G i r i
University of Montreal, Canada
W o r l d S c i e n t i f i c
h Singapore * New Jersey • London • Hong Kong
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Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Fairer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
GROUP INVARIANCE IN S T AT I S T I CAL I NF E RE NCE
Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, orparts thereof, may not be reproduced in anyform or by any
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To
NILIMA
NABANITA
NANDIAN
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CONTENTS
Chapter 0. G R O U P I N V A R I A N C E 1
0,0. Introduction 1
0.1. Examples 3
Chapter I . M A T R I C E S , G R O U P S A N D J A C O B I A N S 7
1.0. Introduction 7
1.1. Matrices 7
1.1.1. Characteristic Roots and Vectors 8
1.1.2. Factorization of Matrices 9
1.1.3. Partitioned Matrices 9
1.2. Groups 10
1.3. Homomorphism, Isomorphism and Direct Product 12
1.4. Topological Transitive Groups 12
1.5. Jacobians 13
Chapter 2. I N V A R I A N C E 16
2.0. Introduction 16
2.1. Invariance of Distributions 16
2.1.1. Transformation of Variable in Abstract Integral 22
2.2. Invariance of Testing Problems 25
2.3. Invariance of Statistical Tests and Maximal Invariant 26
2.4. Some Examples of Maximal Invariants 28
2.5. Distribution of Maximal Invariant 312.5.1. Existence of an Invariant Probability Measure on 0{p)
(Group of p x p Othogonal Matrices) 33
2.6. Applications 33
2.7. The Distribution of a Maximal Invariant in the General Case 36
vii
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viii Contents
2.8. An Important Multivariate Distribution 37
2.9. Almost Invariance, Sufficiency and Invariance 44
2.10. Invariance, Type D and E Regions 45
Chapter 3. E Q U I V A R I A N T E S T I M A T I O N I N C U R V E D
M O D E L S 49
3.1. Best Equivariant Estimation of y, with \ Known 50
3.1.1. Maximum Likelihood Estimators 53
3.2. A Particular Case 53
3.2.1. An Application 58
3.2.2. Maximum Likelihood Estimator 58
3.3. Best Equivariant Estimation in Curved Covariance Models 60
3.3.1. Characterization of Equivariant Estimators of S 613.3.2. Characterization of Equivariant Estimators of 0 63
Chapter 4. S O M E B E S T I N V A R I A N T T E S T S I N
M U L T I N O R M A L S 68
4.0. Introduction 68
4.1. Tests of Mean Vector 68
4.2. The Classification Problem (Two Populations) 75
4.3. Tests of Multiple Correlation 824.4. Tests of Multiple Correlation with Part ial Information 85
Chapter 5. S O M E M I N I M A X T E S T S I N M U L T I N O R M A L E S 91
5.0. Introduction 91
5.1. Locally Minimax Tests 93
5.2. Asymptotically Minimax Tests 106
5.3. Minimax Tests 111
5.3.1. HotelUng's T 2 Test 112
5.3.2. R 2 -Test 124
5.3.3. e-Minimax Test (Linnik, 1966) 135
Chapter 6. L O C A L L Y M I N I M A X T E S T S I N S Y M M E T R I C A L 137
D I S T R I B U T I O N S 137
6.0. Introduction 137
6.1. Eliiptical ly Symmetric Distributions 137
6.2. Locally Minimax Tests in E p(n, S ) 140
6.3. Examples 143
Chapter 7. T Y P E D A N D E R E G I O N S 162
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Group Invariance in
Statistical Inference
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Chapter 0
GROUP INVARIANCE
0 .0 . I n t r o d u c t i o n
One of the unpleasant facts of statistical problems is that they are often
too big or too difficult to admit of prac tica l solutions. Stat istic al decisions
are made on the basis of sample observations. Sample observations often con
tain information which is not relevant to the making of the statistical deci
sion. Some simplifications are introduced by characterizing the decision rules
in terms of the sufficient statistic (minimal) which discard that part of sam
ple observations which is of no value for any decision making concerning the
parameter and thereby reducing the dimension of the sample space to that of
the minimal sufficient statistic. This , however, does not reduce the dimension
of the parametric space. By introducing the group invariance principle and
restricting attention to invariant decision rules a reduction to the dimension
of the parametric space is possible. In view of the fact that sufficiency and
group invariance are both successful in reducing the dimension of the statis
tical problems, one is naturally interested in knowing whether both principles
can be used simultaneously and if so, in what order. Hall, Wijsman and Ghos h
(1965) have shown that under certain conditions this reduction can be carried
out by using both principles simultaneously and the order in which they are
used is immaterial in such cases. However, one can avoid verifying these conditions by replacing the sample space by the space of sufficient statistic and
then using group invariance on the space of sufficient statistic. In this mono
graph we treat multivariate problems only where the reduction in dimension is
1
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2 Group Invariance in Statistical Inference
very significant. In what follows we use the term invar iance to ind icate group
invariance.
In statistics the term invariance is used in the mathematical sense
to denote a property that remains unchanged (invariant) under a group of
transformations. In actual practice many stati stica l problems possess such a
property. As in other branches of applied sciences it is a generally accepted
principle in statistics that if a problem with an unique solution is invariant
under a group of transformations, then the solution should also be invariant
under it. This notion has an old origin in statistical sciences. Apart from this
natural justification for the use of invariant decision rules, the unpublished
work of Hunt and Stein towards the end of Second World War has given this
principle a Strang support as to its applica bili ty and me an in gM ne ss to provevarious optimum properties like min imax, admiss ibil ity etc. of sta tis tical de
cision rules.
Although a great deal has been written concerning this principle in sta
tistical inference, no great amount of literature exists concerning the problem
of discerning whether or not a given statistical problem is actually invariant
under a certain group of transformations. Brillinger (1963) gave necessary and
sufficient conditions that a statistical problem must satisfy in order that it be
invariant under a fairly large class of group of transformations including Lie
groups.
In our treatment in this monograph we treat invariance in the framework of
stat istical decision rules only. D e F i n e t t i (1964) in his theory of exchangeability
treats invariance of the distribution of sample observations under finite permu
tations. It provides a crucial link between his theory of subjective probability
and the frequency approach of probability . T he classica l sta tis tical methods
take as basic a family of distributions, the true distribution of the sample ob
servations is an unknown member of this family about which the statistical
inference is required. Accord ing to De Finett i' s approach no probabili ty is
unknown. If « i , i E s , . . . are the outcomes of a sequence of experiments con
ducted under similar conditions, subject ive uncerta inty is expressed directly
by ascribing to the corresponding random variables X t ,X^, • • • a known joint
distribution. When some of the X's are observed, predictive inference about
others is made by conditioning the orig inal distr ibutions on the observations.De Finetti has shown that these approaches are equivalent when the subjec-
tivist's joint distribution is invariant under finite permutation.
Two other related principles, known in the literature, are the weak invari
ance and the strong invariance principles. T he weak invariance principle is used
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Group Invariance 3
to demonstrate the sufficiency of the classical assumptions associated with the
weak convergence of stable laws (Billingsley, 1968). This is popularly known
as Donsker's theorem (Donsker, 1951). Let X\, X2, • • • be independently dis
tributed random variable with the same mean zero and the variance a 2 and let
Sj = £j=i = ^ . * = | . ;
i
-
1
. - " . « - Donsker proved that { * „ ( ( ) }converges weakly to Bro wni an motion. Strong invariance principle has been
introduced to prove the strong convergence result (Tusnady , 1977). Here the
term invariance is used in the sense that if X i , X j , . . . are independently dis
tributed random variables with the same mean 0 and the same variance a 2
,
and if ft is a continuous function on [0,1| then the limiting distribution of h(Xi)
does not depend on any other property of Xi.
0 .1 . E x a m p l e s
We now give an example to show how the solution to a statistical problem
can be obtained through direct application of group theoretic results.
E x a m p l e 0.1 .1 . Let X a = ( X n l , . . . , X a p ) \ a = l,...,N(> p) be inde
pendently and identically distributed p-variate normal vectors with the same
mean ^ = (m,.. . , j t p ) ' , and the same positive definite covariance matrix E .
The parametric space f! is the space of all {11, £ ) . The problem of testing H0 : // = 0 against the alternatives Hi : 11 0 remains unchanged (invariant)
under the full linear group Gj(p) of p x p nonsingular matrices g transforming
each Xi -* gXi, i = l,...,N. Let
- N N
It is well-known {Giri, 1977) that (X,S) is sufficient for ( / * , £ ) . The induced
transformation on the space of {X,S) is given by
(X,S)^(gX,gSg'), g € Gi(p) • (0.2)
Since this transformation permits arbitrary changes of X, S and any reasonable
statistical test procedure should not depend on any such arbitrary change by
g, we conclude that a reasonable statistical test procedure should depend on(X, S) only through
T 2
= N(N - l l X ' S "1
* . (0.3)
It is well-known that (Giri, 1977) the distribution of T 2 is given by
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4 Group /nuariance in Statistical Inference
fr*(t 2
\6 2
) =(N-l)T(^N-p))
(\ 8
2
y(t 2
j{N -
+
>-l
T{\N + j) if £ 2 > 0 (0.4)
where 6 2
= J V / i ' £ - y Under H 0S 2
= 0 and under ff i S 2 > 0. Apply ing
Neyman and Pearson's Lemma we conclude from (0.4) that the uniformly
most powerful test based on T 2 of H 0 against H i is the well-known Hotelling's
T 2 test, which rejects H 0 for large values of T 2 .
N o t e . In this problem the dimension of fl is p + = P ( P +3
) ) w n i c h is
also the dimension of the (X,S). For the distribution of T2
the parameter isa scalar quantity.
One main reason of the intuitive appeal, that for an invariant problem with
an unique solution, the solution should be invariant, is probably the belief that
there should be a unique way of analysing a collection of statistical data. As a
word of caution we should point out that, if in cases where the use of invariant
decision rule conflicts violently with the desire to make a correct decision or to
have a smaller risk, it must be abandoned. We give below one such example
which is due to Charles Stein as reported by Lehmann (1959, p. 338).
E x a m p l e 0.1.2. Let X = f Xi,...,X p )', Y = (Yi,...,Y p )' be indepen
dently distributed normal p-vectors with the same mean 0 and positive definite
covariance matrices S , 6 S respectively where 6 is an unknown scalar constant.
Consider the problem of testing Ho : 6 — 1 against H\ : 6 > 1, Th i s problem
is invariant under Gj(p) transforming X —> gX,Y —» gY,g e Gt(p). Since this
group is transitive (see Chapter 1) on the space of values of (X, Y) with probability one, the uniformly most powerful invariant test of level a under Gi{p}
is the trivial test $ ( X , Y) = a which rejects Ho with constant probability a
for all values {x,y) of (X,Y). Hence the maximum power that can be achieved
over the alternatives Hi by any invariant test under G;(p) is also a . But the
test which rejects Ha whenever
(0.5)
where the constant C depends on level a, has strictly increasing power 0(6}
whose minimum over the set 6 > Si > 1 is /?{<5i) > /3(1) = a. For more
discussions and results refer to Giri (1983a, 1983b).
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Group Invariance 5
E x e r c i s e s
1. Let Xi,... ,X n be independently and identically distributed normal random
variables with the same mean 9 and the same unknown variance o 2 and let
H A
:d = 0 and H i : 9 / 0.
(a) Find the largest group of transformations which leaves the problem of
testing Ha against Hi invariant.
(b) Using the group theoretic notion show that the two-sided student (-test
is uniformly most powerful among all tests based on (.
2. Univar ia te G e n e r a l L i n e a r Hypothesis . Let , . . . , X n be indepen
dently distributed normal random variables with E(Xi) = (ft, V a r ( X ; ) = a 2 ,
i — 1, . . . , T I . Let fi be the linear coordinate space of dimension of n andlet lip. and I L U be two linear subspaces of fl such that dim I I ^ — k and
dim I I J J — i , I > k. Consider the problem of testing HQ : 8 — ... ,9 P )' G
lTm against the alternatives H i : 9 6 HQ.
(a) Find the largest groups of transformations which leave the problem
invariant.
(b) Using the group theoretic notions show that the usual F-test is uni
formly most powerful for testing HQ against H \ .
3. Let Xy,..., X n be independently distributed normal random variables with
the same mean 9 and the same variance a 2 . For testing H Q : a 2 = a\
against H\ : a 2 = a 2 < af, where <T 2
,O~% are known, find the most powerful
invariant test using the group theoretic notions.
References
1. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
2. D. Brillinger, Necessary and sufficient conditions for a statistical problem to be
invariant under a Lie group, Ann. Math. Statist., 34 (1963) 492-500.
3. B. De Finetti, Studies in Subjective Probability, H. E , Kyburg and H, E . Smoker,
eds., Wiley, New York, (1964) 93-158.
4. M. Donsker, An invariance principle for certain probability limit theorems, Amer.
Math. So c, 6 (1951).
5. N. Giri, Hunt-Stein theorem. Encyclopedia of Statistical Sciences, Vol. 3, Kotz-Johnson, eds., Wiley, New York, 1983a, 686-689.
6. N. Giri, Invariance Concepts in Statistics, Encyclopedia of Statistical Sciences,
Vol. 4, Kotz-Jofmson, eds., Wiley, New York, 1983b, 219-224.
7. N. Giri, Multivariate Statistical Inference, Academic Press, New York, 1977.
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6 Group Invariance m Statistical Inference
8. W. J.Hall, R. A. Wijsman and J . K. Ghosh, The relationship between sufficiency
and invariance with application in sequential analysis, Ann. Math. Statist, 36
(1965) 575-614.
9. E . L. Lehmann, Testing of Statistical Hypothesis, Wiley, New York, 1959.
10. G. Tusnady, in Recent Developments in Statistics, Barra, J. R., Brodeau,
F. , Romier, G. and Van Cutsera, B. eds, North-Holland, Amsterdam, (1977)
289-300.
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Chapter 1
MATRICES, GROUPS AND JACOBIANS
1 . 0 . I n tr o d u c t i o n
The study of group invariance requires knowledge of matrix algebra and
group theory. We present here some basic results on matrices, groups and
Jacobians without proofs.
Proofs can be obtained from Gir i (1993, 1996) or any textbook on these
topics.
1 .1 . Matr ice s
A p x q matrix C = (c \j) is a rectangular array of real number Cy written
as
where e% is the element in the it h row and the jth column. The transpose C
of C in a q x p matrix obtained by interchanging the rows and the columns of
C. If q = p, C is called a square matrix of order p. K q — 1,0 is a 1 X p row
vector and if p = 1, C is a g x 1 column vector.
A square matrix C is symmetric if C — C.
A square matrix C = (c ^) of order p is a diagonal matrix D ( c u , c pp )
with diagonal elements e n , . . . ,Cp P if all off-diagonal elements of C are zero.
A diagonal matrix with unit diagonal elements is an indentity matrix and
is denoted by / . Sometimes it will be necessary to write it as I K
to denote an
identity matrix of order k.
(1.1)
7
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8 Group Invariance in Statistical Inference
A square matrix C — (cij) of order p is a lower triangular matrix if =
0 for j > i. The determinant of the lower triangular matrix det C = H(=i c » -
We shall also write det C — \C\ for convenience.
A square matrix C = (ey ) of order p is a upper triangular matrix if Cj, = 0
for i > j and det C = [IS=Ic
"-A square matrix of order p is nonsingular if det C ?^ 0. If det C = 0 then
C , is a singular matrix.
A nonsingular matrix C of order p is orthogonal if CC = CC = I .
The inverse of a nonsingular matrix C of order p is the unique matrix C _ 1
such that C C - 1 = C _ 1 C = J . From this it follows that det C " 1 = (det C)~ l .
A square matrix (7 = (c^ ) of order p or the associated quadratic form
x'Cx = J2i Z l j djXiXj is positive definite if x'Cx > 0 for x = ( i i , . . . , x p )' / 0.
If C is positive definite C _ 1 is positive definite and for any nonsingular matrix
A of order pACA' is also positive definite.
1.1.1. Characteristic roots and vectors
The characteristic roots of a square matrix C of order p are given by the
roots of the characteristic equation
d e t ( C - A / ) = 0 (1.2)
where A is real. A s
det {8C8' - XI ) = det ( C - XI )
for any orthogonal matrix 8 of order p, the characteristic roots of C remain in
variant under the transformation of C —> 8C8'. The vector x — ( i ^ , . . . , x p )' /
0 satisfying{C -X I )x = 0 (1.3)
is the characteristic vector of C corresponding to its characteristic root X. If x
is a characteristic vector of C corresponding to its characteristic root X, then
any scalar multiple ax, a j i 0, is also a characteristic vector of C corresponding
to X.
S o m e R e s u l t s o n C h a r a c t e r i s t i c R o o t s a n d V e c t o r s
1. T h e characteristic roots of a real symmetric matrix are real.
2. The characteristic vectors corresponding to distinct characteristic roots of
a symmetric matrix are orthogonal.
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Matrices, Groups and Jacobians 9
3. T he character ist ic roots of a symmetri c positive definite matrix C are ail
positive.
4. Given any real square symmetric matrix C of order p, there exists an orthog
onal matrix 9 of order p such that 9C9' is a diagonal matrix D{\\,..., A p )
where A i , . . . , A p are the characteristic roots of C. Hence det C — n f = i A<
and tr C — X]f=i *W- Note that tr C is the sum of diagonal elements of C.
1.1.2. Factorization of matrices
In this sequal we shall use frequently the following factorizations of
matrices.
For every positive definite matrix C of order p there exists a nonsingular
matrix A of order p such that C = AA' and, hence, there exists a nonsingular
matrix B (B = A'1 ) of order p such that BCB' = I.
Given a symmetric nonsingular matrix C of order p, there exists a nonsin
gular matrix A of order p such that
- i ) m
where the order of / is equal to the number of positive characteris tic roots of C and the order of —I is equal to the number of negative characteristic roots
of C .
Given a symmetric positive definite matrix C of order p there exists a
nonsingular lower traingular matrix A (an upper triangular matrix B) of the
same order p such that
C = AA' = B'B. (1.5)
C h o l e s k y D e c o m p o s i t i o n For every positive definite matrix C there exists an unique lower triangular matrix of positive diagonal elements D such that
C = DD'.
1.1.3. Partitioned matrices
Let C — ( c y) be a p x q matrix and let C be partitioned into submatrices
Cij as
C = ( ^ n
^ \ C a i C22 /
where C u - (cij)(i = l , . . . , m ; j = l,...,n)\C 12 - (ftj)(» = l , . . . , m ; jf =
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10 Group Invariance in Statistical Inference
n+ l , . . . , q ) ; C 2 l = (cy)(t = m+ l , . . . , p ; j = l , . . . , n ) ; C 2 2 = =
m + l j . . , ,p; 3' = J l + 1 , . . . . , j j j . For any square matrix
C — f^1' \
\C2i C22 / where Cn,C 2 2 are square matrices and C 2 2 is nonsingular
(1) det ( C ) = det ( C 2 2 ) det ( C n - C a C ^ C j i ) ,
(2) C is positive definite if and only if C U ,C 22 - C 2iG^C- n or equivalently
C22-.Cn — C\ 2C 22C 2 \ are positive definite.
L e t C - 1 = B be simi lar ly partitioned into submatrices Bi:i ,i,j = 1,2.
Then
C^,1
— Bu - B12B22B21, C22 — B22 - B 2iBll 'Bi 2 ,
C^C 12 = -Bl2 B^. (1.6)
1.2 . Groups
A group G is a set with an operation r satisfying the following axioms.
A i . For any two elements a, b E G, arb € G.
A 2 . For any three elements a,b,c G G\ {arb)rc — or(brc) .
A3. There exists an element e (identity element) such that for all o £ G,are—
a.
A 4 . For any a G G there exists an element a~ l (inverse element) such that
a r a - 1
— e.
In what follows we will write for the convenience of notation arb = ab. In
such writing the reader may not confuse it with the arithmetic product ab.
A group G is abelian if for any pair of elements o, 6 belonging to G, ab = ba.
A non-empty subset H of G is a subgroup if the restriction of the group
operation T to H satisfies the axioms A i , , . . , A4,
E x a m p l e s of G r o u p s
E x a m p l e 1.1. A. Le t X be a set and G be the set of all one-to-one
mappings
g : X -> X
with g{x) = g(y)\x,y G I ; implies x — y and for x G X there exists y G X such
that y = g{x). With the group operation defined by
9r92(x) =SI(£J2M);SI,S2 € G ,
G forms a permutation group.
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Matrices, Groups and Jacobians 11
E x a m p l e 1.2. The additive group of real numbers is the set of all reals
with the group operation ab = a + 6. The multiplicative group of all nonzero
reals with the group operation ab — a multiplied by b.
E x a m p l e 1.3. Let X be a linear space of dimension n.
Define for x 0 G X,g xo(x) — x + X Q,X G X. T h e set of a ll {g Xo } forms an
additive abelian group and is called the translation group.
E x a m p l e 1.4. Let X be a linear space of dimension n and let Gi(n) be
the set of all nonsingular linear transformations X onto X. Gi(n) with matrix
multiplication as the group operation is called the full linear group.
E x a m p l e 1.5. The affine group is the set of pairs (g,x),x G X,g € Gi{n)
with the group operation defined by (9i,£i)(<?2,X2) = {9192,91X2 + Xi) with
91.92 6 Gi(n) and X\,x 2 G X. T h e identity element and the inverses are
E x a m p l e 1.6 . T h e set of all nonsingular lower triangular matrices of
order p with the usual matrix multiplication as the group operation forms
a group GT(P) (with identity matrix as the unit element). The set of allupper triangular nonsingular matrices of order p forms the group GUT(P) with
identity matrix as the unit element.
E x a m p l e 1.7 . T h e set of all orthogonal matrices 6 of order p with the
matrix multiplication as the group operation forms a group 0(p).
E x a m p l e 1.8. Let o C £1 C £2 C • • • C X k = X be a strictly increasing
sequence of linear subspaces of X and let G be a subgroup of Gi(p) such that
g G Gi{p) if and only if gXi = Xi, i = 1 , . . . , k. T h e group G is the group of
nonsingular linear transformations on X to X which leaves Xi invariant.
Choose a basis x^\ ... , x $ , t = 1 , . . . , k for Xi-Xi-i with n, — dim (X,) -
dim (3t,_i) and XQ — <f> (null set).
Then g G G can be written as
(1,0) and ( s . x r 1 - O r 1
9 =
/ 9(U) 9(12)0 9(22)
fl(lJfc) \ 9<2fc)
(1.7)
\ 0 0
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12 Group /nuariance in Statistical inference
where g/*^ is a block of Bj rows and rij columns for i < j . I f n; = 1 for al l
i,G = GUT-
1.3. H o m o m o r p h i s m , I s o m o r p h i s m a n d D i r e c t P r o d u c t
Let G, H be groups. A mapping / of G into H is called a homomorphism
if, for gi, g 2 £ G ,
/(9i!72) = / ( 9 i ) * / M (1-8)
where * is the group operation in H. I f e is the identity element in G , }{e)
is the identity element in H and / ( g - 1 ) — [/( <?) ] - 1 . I n addition, if / is an
one-to-one mapping, then / is called an isomorphisim.
The Cartesian product G x H of groups G and H with the operation(9u hiK92,h 2 ) = (9i92,hih 2 y,gi,g2 £ GM M e H is called the direct
product of G and if.
A subgroup H of G is a normal subgroup of G if for all h £ H and s e (?,
ghg~l £ / / , or equivalently if gHg
-1 — H .
E x a m p l e 1.9. The group Gf ( p ) of lower triangular matrices of order p
with positive diagonal elements is a normal subgroup of GT ( P)- Any subgroup
of an abelian group is normal.
Let G be a group and let i f be a subgroup of G. The set G /H of all sets
of the form gH, g € G is called the quotient group of G modulo H with group
operation defined by, for j&jjfe G &,.(%H){gzH) is aset of elements obtained
by multiplying all elements of giH with all elements of g 2 H. The identity
element is H and (gH)-1
— g~v
H.
1.4. Topological T r a n s i t i v e G r o u p s
Let X be a set and A a collection of subsets of X. (X,A) is called a
topological space if A satisfies the following axioms:
TAi : X £ A.
T A j : The union of an arbitrary subfamily of A belongs to A.
T A 3 ; The intersection of a finite subfamily of A belongs to A.
The elements of A are called open sets. If, in addition the topological space
(X,A) satisfies: for x,y £ X,x / y, there exist two open sets A,B, belongingto A such that A n B = <p with x e A, y £ B; then it is called a Hausdorff
space.
A collection of open sets is a base for a topology if every open set in A is
the union of a sub-collection of sets in the base.
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Matrices, Groups and Jacotians 13
If a group G has a topology r defined on it such that under T the mapping
ab from G x G into G and a - 1 from G into G ( a , b £ G ) are continuous, then
( G , r ) is a topological group.
A compact group is a topological group which is a compact space. A loca lly
compact group is a topological group whi ch is a locally compact space.A compact space is a Hausdorff space X such that every open covering of X
can be reduced to a finite subcovering. A locally compact space is a Hausdorff
space such that every point has at least one compact neighborhood.
Let X be a set. T h e group G operates from the left on X if there exists a
function on G x X into X whose value at {g,x} is denoted by gx such that (i)
ex — x for all x € X and e, the identity element of G ; (i i) g^igix) — gzgi(x).
This implies that gtG is one to one on X into X.
Let G operate from the left on X. G operat es t ran s i t ive ly on X if for
every Xi,x 2 £ X there exists ag £ G such that gx\ — x 2-
E x a m p l e 1.10. Let X be a linear space. T he full linear group operates
transitively on X — { 0 } .
E x a m p l e 1.1 1. T h e group 0{n) of n x n orthogonal matrices operates
transitively on the set of all n x p(n ^ p) real matrices, x satisfying x'x =
E x a m p l e 1.1 2. T h e linear group G; (p ) acts transitive ly on the set of all
p x p positive definite matrices s, transfering s to gsg',g £ G/(p) .
1 .5 . Jacob ians
Let Xi,...,X n be a sequence of continuous random variables with pdf
fx, xAXi,--.,X n
} and let Yi = gi(X u
... ,X n
)>i = l,...,n be a set of continuous one-to-one transformations of X%, • • •, X n^ Assume that the func
tion gi,...,g n have continuous partial derivatives with respect to X ] . . . , X n .
Denote the inverse functions by X ; = hi(Yi,..., Y n ),i = l , . . . , n . Let J - 1 be
the determinant of the n x n matrix of partial derivatives
J is called the Jacob ian of the transformation X\,..., X n to V j , . . . , Y n .
The pdf of Yi,..., Y n is given by
M i M i
9Y„
(1.9)
dX„9V„
M, . . . r y-(Ki.--- . Vn) = fx,,...^Cftltflj • • • . » » ) . . • • * M i * l . - - v . S n ) | / |
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14 Group Invariance tn Statistical Inference
where \J\ is the absolute value of J. We now state some results on Jacobian
without proof. We refer to Gir i (1994), OIkin (1962) and Roy (1959) for further
results and proofs.
Some R e s u l t s o n J a c o b i a n s
Let X = ( X i , . . . , X p ) ' , y = ( Y 1 , . . . , Y P ) ' € Ev . The Jacobian of the
transformation X —> Y — gX g g G j( p) is (det g)~l
•
Let X, Y be p x n matrices. T h e Jacobian of the transform ation X —* Y =
gx,geGt{p) is (det g)"1 .
Let X,Y bepxn matrices and let A 6 Gj(p),Z? € Gt{n). The Jacobian of
the transformation X -> Y = AXB is (det A'1
)* (det S -l
)p
.
Let g, k € G r ( p ) . The Jacobain of the transformation g —» kg is
n f = i ( f t . i ) _ 1 where h — (/i ,j) and the Jacobian of the transformation g —* gh is
n f = 1 < M ' - p - 1 .
Let g,k e GUT ( P)- T h e Jac obian of the transform ation g —» kg is
n ^ i C l » ) ' - p - 1 where k = (hij) and the Jacobian of the transformation g —• gh
^ n L
Let GST - (P ) be the multiplicative group of p x p lower tria ngu lar nonsingular
matrices in the block form, i.e. g e G B T ( P ) .
9(21) 9(22) 0
•9(1:1) 9(*2) * • • • 9(ltfc)
9 = \ m m 0 |
where g^q are submatrices of order pi x pi such that Y^iPi = P- F ° r 9.^ 6
GBT ( P) the Jacobian of the transformation g -* kg is n ! = i i d e t / i ( i i ) j - , T i ,
where cr; = Y? 3=\P3* a
o=
°- T h e Jacob ian of the transformation g —> gh
is n t j d e t h ^ F -1
"p
.
Let GBUT ( P) be the group of nonsingular upper triangular matrices of order
p in the block form i.e. g e GBUT,
' 9 (H ) 9(12) • • • 9<u) \
? _ I 0 9(22) •• 9(2fc) j
0 0 9(fcfc)
where g^ are submatrices of order p; satisfying P i = p . For g,h e G e t 7 T ,
the Jacobian of the transformation g -t gh is n L i [ d e t fya)]-"' a n d t h a t o f
9 ^ / » 9 i s n L i [ d e t « ( u ) ] " i - , " p .
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Matrices, Groups and Jacobians 15
Let 5 be a symmetric positive definite matrix of order p. The Jacobian of
the transformation 5 -> gSg',g e Gt(p) is [det g ] " t p + 1 ) and that of S - » 5 _ 1
is (det S ) 2 " .
Let 5 be a symmetric positive definite matrix of order p and let g be the
unique lower triangular matrix such that S = gg'. The Jacobian of the transformation S -> gSg' is 2~" ^ i f f f i i ) - ^ 1 - * 5 with g = (jfc), The Jacobian of
the transformation S —• gSg',g g G £ ( p ) is [det g]~<
- p+1
K
E x e r c i s e s
1. Show that for any nonsingular symmetric matrix A of order p and non null
p vectors x,y
(c) |A + xa;'| = \A\{\ + x'A~1
x).
2. Let A,B be p x q and qxp matrices respectively. Show that \I P + AB\ =
\I q+BA\.
3. Show that for any lower triangular matrix C the diagonal elements are its
characteristic roots.
4. Let X be the set of all n x p real matrices X satisfying X'X — I p . Show
that the group 0(n) of orthogonal matrices 9 of order p, which transform
X —> 6X acts transitively on X.
5. Let 5 be the set of all symmetric positive definite matrices s of order p.
Show that G j( p) which transforms s —t gsg',g £ G)(p), acts transitively on
5.
References
1. N. Giri, Multivariate Statistical Inference, Academic Press, New York, 1977.
2. N. Giri, Introduction to Probability and Statistics, 2nd Edition (Expanded), Marcel
Dekker, New York, 1993.
3. N, Giri, Multivariate Statistical Analysis, Marcel Dekker, New York, 1996.
4. I. Olkin, Note on the Jacobian of certain transformations useful in multivariate analysis, Biometrika, 40 (1962), 43-46.
5. S. N. Roy, Some Aspects of Multivariate Analysis, Wiley, New York, 1957.
(b) a?{A + xx')-1
x =
(a) (A + xy')-1
^l + y'A~
l
x
x'A-^x
l + x'A-tx1
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Chapter 2
INVARIANCE
2 . 0 , I n t r o d u c t i o n
Invariance is a mathematical term for symmetry. In prac tice many sta
tistical problems involving testing of hypothesis, exhibit symmetries, which
imposes additional restrictions for the choice of appropriate statistical tests,
for example, the statistical tests must also exhibit the same kind of symmetry
as is present in the problem. In this chapter we shall consider the principle of invariance of statistical testing problems only. Fo r an addit ional reference the
reader is referred to Lehmann (1959), Ferguson (1967) and Nachbin (1965).
2 .1. In va r ia nce o f D i s t r ib ut io ns
Let [X,A) be a measure space and ft the set of points 8, that is, f! — {9}.
Consider a function P on ft to the set of probability measures on (X t A) whose
value at 8 is P B
. Let G be a group of transformations operating from the lefton X such that g € G
g : X°"°X is (A, A) measurable; (2.1)
and
9 : ft ° - 0 (2.2)
is such that if X has distribution P6 , gX, X € X has distribution P s- where
0* = 98 6 ft. Al l transformations considered in connection wi th invar iance
will be taken for granted as one-to-one from X onto X. An equivalent way of
stating (2.2) is as follows:
16
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Invariance I 7
Pe{gX e A) = P- g6 (X €A),Ae A, (2.2a)
or
P 9{g-1
A) = P se{A) (2.2b)
This can also be written as
Pt(B) = P s»{gB), Be A
If all Pe are distinct, that is, 81 / 82 Pe, PH * t n e n
9 * S a homomorphism.
The condition (2.2a) is often known as the condition of invariance of the dis
tribution.
Very often in statistical problems there exists a measure A such that Pa
is absolutely continuous with respect to A for all 8 S fi so that p$ is the
corresponding probability density function with respect to the measure A. I n
other words, we can write
Also in great many cases of interest, it is possible to choose the measure A
such that it is invariant under G, viz
Defin i t ion 2 .1 .1 . A measure A satisfying (2.4) is left invariant.
In such cases the condition of invariance of distribution under G reduces to
The general theory of invariant measures in a large class of topological
groups was first given by Haar (1933). However, the results we need were
all known by the end of the nineteenth century. In the terminology of Haar,
Ps(A) is called a positive integral of p«, not necessarily a probability density
function. The basic result is that for a large class of topological groups there is
a left invariant measure, positive on open sets and finite on compact sets and
to within multiplication by a positive constant this measure is unique. Haar
proved this result for locally compact topological groups. T h e definition of
right invariant positive integrals, viz. Pe{A) = Pge(Ag) is analagous to that
for left invariant posit ive integrals.
(2.3)
\(A) = \[gA) for all A G A, geG. (2.4)
Pgs(gx) = pe(x) for alia; e X, g e G. (2.5)
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18 Group Invariance in Statistical Inference
Because of the poineering works of Haar such invariant measures are called
invariant (left or right) Haar measures. A rigorous presentation of Haar mea
sure will not be attempted here. The reader is referred to Nachbin (1965).
It has been shown that such invariant measures do exist , and from a left
invariant Haar measure one can construct the right invariant Haar measure.
We need also the following measure-theoritic notions for further develop
ments. Let G be a locally compact group and let B be the <7-algebra of compact
subsets of G.
Def in i t ion 2.1.1. (Relatively left invariant measure). A measure v on
(G,B) is relat ively left invariant with left multiplier x{g) i f f satisfies
v{gB) = x(9)v(B), f o r a l l B e B . (2.5a)
The multiplier x(t») is a continuous homomorphisim from G —» R+ . I f v
is relatively left invariant with left multiplier \(g) then \ ( g - 1 ) = l/x(ff) and
x(ff - l )f(^ff) ' s a l eft invariant measure.
Def in i t ion 2.1.2. A group G acts topologically on the left of x if the
mapping T(g, x) : x x G — » x
1S
continuous.
Def in i t ion 2.1.3. (Proper G-space) . Le t the group G acts topologically
on the left of x a n d let h be a mapping on G x \ — * X x X given by
Hg>x) = {gx,x)
for g G G,x € X- The group G acts properly on \ if for every compact
C C x x X> / »" J {G) is compact.
The space x is called a proper G-space fG acts properly on \ . A n equivalent
concept of the proper G-space is the Cartan G-space.
Def in i t ion 2.1.4 (Cartan G-s pace). Let G acts topologically on \ . Then
X is called a Cartan G-space if for every x e x there exists a neighborhood V
of a such that (V, V ) = {g e G\{gV) n V # 4>} has a compact closure.
Wijsman (1967, 1978,1985, 1986) demonstrate the verif ication of the condi
tion of Cartan G-space and proper actions in a number of multivariate testingproblems on covariance structures.
E x a m p l e 2.1.1. Let G be a subgroup of the permutation group of a finite
se t* - For anys ubset A of x define A(A) = number of points in A. Themeasure
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Invariance 19
A is an invariant measure under G and is unique upto a positive constant. It
is known as counting measure.
E x a m p l e 2.1.2. Let x be the Euclidean n space and G the group of
translations defined by i3
e % t
g Xl
e G implying
g Xl (x) =x + xi.
Clearly G acts transitively on x- T h e n-dimens ional Lebesgue measure A is
invariant under G. Again A is unique upto a positive multiplicative constant.
E x a m p l e 2. 1. 3. Consider the group Gi(n) operating on itself from the
left. Since an element g operating from the left on Gi(n) is linear, it has aJacobian with respect to this linear operation. Obvious ly, Gi(n) is a vector
space of dimension n 2 . Let us make the convention to display a matrix of
dimension n x n b y column vectors of dimension n x 1, by writing the first
column horizontally. Then the second is likewise and so forth. In this manner
a matrix of dimension nxn represents a point in R n . L et g,x,y e Gi(n) and
let g = (gi } ),x — (x z j),y — (y t] ). If we display these matrices by their column
vectors, the relationship y — gx gives us the coordinates of the point
as
Vij = y^9ikXkj
k=l
in terms of the coordinates of
{^111 • • ' *^ l n > "' • ! X ni, . . . , Xjin) -
The Jacobian is thus the determinant of the n 2 x n 2 matrix
(9 0 . . . 0^
0 g ... 0
\ 0 0 9>
where each 0 indicates null matrix of dimension nxn. Hence the Jacobian of
the transformation y = gx is (detg} n
. Let us now define for y e Gt{n)
dX(y) -(det T / ) "
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20 Group Invariance in Statistical Inference
Then
dX(gy) =(det g) nY[dVij
(det(gy))"
(det 3/)"
= dX(y)
Hence A is an invariant measure on G ;(n) under (?/(n). It is also unique upto
a positive multiplicative constant.
E x a m p l e 2.1.4. Let G be a group o f n x n nonsingular lower triangular
matrices with positive diagonal elements and let G operate from the left on
itself. We will verify that the Jacobian of the transformation
Q-*hg
for G, is I l i C1
" ) ' - Thus an invariant measure on G under G which is
unique upto a positive multiplicative constant , is
dX{g) =i l > ,
n?= i ( j ? » ) '
To show that the Jacobian is n r = i (^ « )*> ' e t 9 - ( f i j ) , ^ = with g tj =
kij = 0 for i <j. Then
This defines a transformation
( f t i ) - ( C y ) .
We can write the Jacobian of this transformation as the determinant of the
matrix /dC n dCu 3 C „ \
dglL &921
dC 21 dC 2l
dC 21
dgu dgnn
BC nn dC nn
\ dgn $921 &9nn 1
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/nuartonce 21
which is a [n(rc +1) ] /2 x [n( fi -r l) ]/ 2 lower triangular matrix with the diagonal
elements
- h kk ,k > ro.
In other words h%i will appear once as a diagonal element, /122 will appear
twice as a diagonal element and finally h nn will appear n times as a diagonal
element. Hence the Jacobian is I I I L i C1
" )1
'
E x a m p l e 2.1.5. Let S be the space of all positive definite symmetric
matrices S of dimension n x n . and let Gi(rc) operate on 5 as
S^gSg', SeS, g 6 G,{n). (2.6)
To show that the Jacobian of this transformation is (det g) n+1 , let i?y be the
matrix obtained from the nxn identity matrix by interchanging the ith and
the jth row; M^C) be the matrix obtained from the n x n identity matrix by
multiplying the ith row by any non-zero constant C and let Ay be the matrix
obtained from the nxn identity matrix by adding the jth row to the ith. We
need only show this for the matrices Eij,Mi(C) and Ay. This can be easily
verified by the reader; for example, det(M,(C)) = C and Mi{C)SMi(C) is
obtained from S = (5y) by multiplying Sa by C 2 and Sy for i ^ j by C so
that the Jacobian is C ^ " "1 = C "
+ 1
.
Let us define the measure X on 5 by
d X { b }
- ( d e t S ) ( « W
Clearly,
d\(S) = dX(gSg'),
so that X is an invariant measure on S under the transformation (2.6).
Here also X is unique upto a positive mult iplicative constant. T h e group
G;(p) acts transitively on S.
E x a m p l e 2.1.6. Let H be a subgroup of G. The group G acts transitively
on the space x = G/H (the quotient group of H) where group action satisfies
9i (gh) = 9 i9H i for 9i e G , g€G.
When the group G acts transitively on x, there is a one to one correspondence
between x and the quotient space G/H of any subgroup H of G.
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22 Group Invariance in Statistical Inference
2.1.1. Transformation of variable in abstract integral
Let (X, A,n) be a measure space and let (y, C) be a measurable space.
Suppose <p is a measurable function on X into y. Let us define the measure v
on (y,C) by
v(C) = ti{>p-1(C)), CEC. (2.7)
T h e o r e m 2.1.1. If f is a real measurable function on X such that f{x) —
g[tp(x)) and if f is integrable, then
I }{x)dti{x) = f g(<p(x))dn(x)
J x Jx
= f g{y)dv{y) (2.8) Jy
For a proof of this theorem, the reader is referred to Lehmann [(1959), p. 38],
E x a m p l e 2.1.6. ( W i s h a r t d i s t r i b u t i o n ) . L e t X\,..., X n be n indepen
dent and identically distr ibuted normal p-vectors wi th mean 0 and covariance
matrix £ (positive definite). Assume n > p so that
is positive definite almost everywhere. T he join t probabil ity density function
of X 1} ... ,X n is given by
p ( x 1 , . . . , x J - ( 2 7 r ) - " P / 2 ( d e t £ - 1 ) " / 2 x e x p | - i t r S - 1 ^ ^ ; | . (2.9)
Using Theorem 2.1.1, we get for any measurable set A in the space of S
P(S e A) — ( 2 J T ) - ' 1 ^ 2
J (det £ _ 1 ) n / 2
s - X^Xixl e A exp j - i t r S ^ s j f[dx,, (2.10)
' i=l
= (2^)-^ (det £ _ 1 ) n / 2 exp j - i tr dm(s)
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Invariance 23
where m is the measure corresponding to the measure v of Theorem 2.1.1. Let
us define the measure m' by
To find the distribution of S, which is popularly called Wishart distribution
( W ( S , J I ) ) with parameter E and degrees of freedom n, it is sufficient to find
dm'. To do this let us first observe the following:
(i) Since E is positive definite symmetric, there exists a € Gi{p) such that
£ — aa'.
(ii) Let
As a 1 Xi , s are independently normally distr ibuted with mean vector 0 and
covariance matrix I (identity matrix), 5 is distributed as W(I,n). Thus
P aa .(S 6 A) = ( 2 7 r } - n ^ 2 / (det(aa')-l s)) n/2
JatA
x exp | - ^ t r ( a a ' ) - 1 s | dm'(s);
P r(aSa' € A) = ( 27 r ) - n ^ 2 / (de t ( ( c .a ' ) _ 1 s))/a
x exp j - i t r ( a a ' )l
s | x d m * { aJ
5a ' )
Since P a a ' ( S e A) = Pj{aSa € A) for all measurable sets A in the space of
5, we must have for all g £ GJ ( J I )
dm'(s) = dm t(gsg'). (2.12)
By example 2.1.5 we get
d m { S ) = (det '
where C is a posit ive constant.
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24 Group Invariance in Statistical Inference
Thus the probability density function of the Wishart random variable is
given by
W ( E , n ) = C(det £ - 1 } n ' 2 ( d e t s ) ( n - p - , ) / 2 exp j - ^ t r £ _ 1 s j (2. 13)
where C is the universal constant depending on E . To evaluate C let us first
observe that C is a function of n and p. Let us write it as C = C n , p . Since C n ̂p
is a constant for all values of £ , we can, in particular, take E = J to evaluate
C np . Write
Sn S12
\ $2l S22
where S u is (p — 1) X (p — 1). Make the transformations
£11 = Tn ,
S12 — T12,
S22 - SuSiiSu = T22 = Z ( s a y ) ,
which has a Jacobian equal to unity.
Using the fact that
\ S\ = \Su\ \S22 - S 2 iSi,l
Si2\ ,
J C n , p | 5 | l n - p - 1 » ^ e x p | - ^ t r 5 j d s = 1 f ora l l p .n ,
J C n , p \S \^-"-1>' 2e^p^~tis^ds
=C
-.pj k i i l | n " p " 1 ) / 2 e x p | - i t r S l l J d 5 l l
x y " { , ) ( - p - i ) / 2 e x p | _ I z | d 2
* / e x p { ~ 5 s ' i 2 s i " i 1 ' 5 l 2 } d l l » 2
and
we get
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/nuor iance 25
I x / | s n |
t n
"p
~l ) / 2
e x p da11
m C n , P ^ 2 j r j ( P - i V 2 2 ( n - P + n / 2 r f n - p + l \C n,(p-1) \ 2 J
j C " , ( p - i ) | 5 n | ( n ^ l / 2 e x p dan •
Since C n , i = 2- n' 2 /T{n/2), we have
7 r - p ( p - l ) / 2 2 - ' 1 / 2 J r - p ( p - l ) / 2 2 - n p / 2
2.2. I n v a r i a n c e of T e s t i n g P r o b l e m s
We have seen in the previous section that for the invariance of the statistical
distribution, transformation g of g g G must satisfy g9 g fi for 9 g fl .
Def in i t ion 2.2.1. ( I n v a r i a n c e of pa ra me tr ic sp ac e) . The parametric
space fi remains invariant under a group of transformation G : X A1
, if
(i) for g g G, the induced mapping j on fi satisfies g9e.il for all # € fi;
(ii) for any 8' £ fi there exists 0 g fi such that g$ = 8'.
An equivalent way of writing (i) and (ii) is
fffi = fi. (2.14)
It may be remarked that if P g for different values of 8 g fi are distinct then
g is one-to-one. T he following theorem will assert that the set of all induced
transformations g,g g G also forms a group.
T h e o r e m 2.2.1. Letgly g 2 be two transformations which leave fi invariant.
The transformations g 2 gi and g^1
defined by
0201 (x) = 02(9i (aO)_ . (2-15)
ffi 9iW=x
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26 Group /rtuariance in Statistical Inference
for all x £ X leave SI invariant and g 2 gi — fftffi, (Si l ) — (ffi) 1
Pr oo f . We know that if the distr ibution of X is Pg, 0 G fl, the distribution
of gX is Pgd.gi 1 € ft. Hence the distribution of g2gi {X) is P g ^ e , as <hf) € ft
and leaves fi invariant, g~2g ~\6 € ft. Thus g 2 g\ leaves ft invariant and,
obviously,
9i gi = 92fli •
Similarly the reader may find it ins tru ctive to verify the other assertion . •
Let us now consider the problem of testing the hypothesis Ho : 6 G Slu0
against the alternatives Hi : 6 £ ftjj,, where ft#0 and Sin, are two disjoint
subsets of SI. Let G be a group of transformations which operate from theleft on X satisfying conditions (i) and (ii) above and (2.14). I n what follows
we will assume that g £ G is measurable which ensures that whenever X is a
random variable then gX is also a random variable.
Def ini t i on 2 .2 .2. ( I nv ar ia nc e of s ta t i s t i c a l p r ob le m ). T he problem
of testing Ho : 8 £ SIH U
against the altern ative H i : 0 <S Sin, remains invariant
under a group of transformations G operating from the left on X if
(i ) (or g € G, A € A, Pe(A) = P- sa(gA),
(ii) = ttH a ,gSl Hl = Sl Hl for all g G G .
2 . 3. I n v a r i a n c e o f S t a t i s t i c a l T e s t s a n d M a x i m a l I n v a r i a n t
Let (X,A, A) be a measure space. Suppose pg,8 G SI, is the probabilitydens ity function of the random variable X G X with respect to the measure
A. Consider a group G operating on X from the left and suppose that the
measure A is invariant wit h respect to G , that is,
\{gA) = \(A) f o r a l l A e A ? € G .
Let [y,C) be a measurable space and suppose T ( X ) is a measurable map
ping from X into X.
De fi ni t io n 2 .3 .1 . (I n va ri an t fu nc ti on ). T ( X ) is an invariant function
on X under G operating from the left on X if T ( X ) = T(gX) for all g G
G,X€X.
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Invariance 27
De f i n i t i on 2 . 3 . 2 . ( M a x i m a l i n v a r i a n t ) . T(X) is a maximal invariant
function on X under G if T(X) is invariant under G and T(X) = T{Y) for all
X. Y £ X implies that there exists g £ G such that Y = gX.
If a problem of testing HQ : 8 e fl# 0 against the alternatives H lt8 £ fijjr
remains invariant under the group transformations G, it is natural to restrictattention to statistical tests which are also invariant under G. Let v>(X), X £ X
be a statistical test for testing Ho against H i , that is, <p{x) be the probability
of rejecting H a when x is the observed sample point. If f is invariant under G
then ip must satisfy
<p(x) = tp(gx), xeX.geG (2.16)
A useful characterization of invariant test f(X) in terms of the maximal in
variant T(X) on X is given in the following theorem.
T h e o r e m 2.3.1. Let T(X) be a maximal invariant on X under G. A test
ip(X) is invariant under the group G if and only if there exists a function h
for which <p(X) = h(T{X)).
P r oof . Let <p(X) = h{T(X)). Obviously,
<p(x) = <f(g(x)), g£G,x£X.
Conversely, if <p(x) is invariant and if T{x) — T(y);x,y £ X then there exists
a g £ G such that y — gx and therefore tp(x) — f[y). •
In general if ip is invariant and measureable, then <fi(x) — k(T(x)) but h
may not be a Borel measurable function. However if the range of the ma xima l
invariant T(X) is Euclidean and T is Borel measurable then Ai — {A : A £
A and gA — A for all g £ G) is the smallest f-field with respect to which T is
measurable and ip is invariant if and only if ip(x) — h(T(x)) where h is a Borel
measurable function. Thi s result is essentially due to Blackwell (1956).
Let G be the group of induced transformations on f! (induced by the group
G). Let v(8),8 € f i, be a maximal invariant on fi under G.
T h e o r e m 2.3.2 . The distribution of T(X) [or any function of T(X)) depends on f! only through i/{9).
Proof. Suppose v{9i) = v(8' 2 ),8\,8i £ fi. Then there exists a g £ G such
that f?2 — § # i . Now for any measurable set C
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28 Group Invariance in Statistical Inference
P tl (T(X) £ C) = P 9i{T{gX)) G C)
= P- g9l (T(X) E C )
•
2 .4 . S o m e E x a m p l e s o f M a x i m a l I n v a r i a n t s
E x a m p l e 2.4.1. Let X = {Xu • - • . * n ) ' £ * and let G be the group of
translations gc{X) = (Xi + C,.. .,X a + C)',-oo < C < oo.
A maximal invariant on X under G is T{X) = (X\,—X n ,.. — X n ).
Obviously, it is invariant. Suppose x — (x\,... , £ „ ) ' , y = ( j / i , . . . ,y n )' and let
T(x) = T(y). Writing C — y n—
x n we get yi = Xi + C for i—
1 , . . . , n.
Another maxim al invariant is the redundant (X\ — X,..., X n — X) where
X = i 51™ X j . It may be further observed that if n — 1 there is no maximal
invariant. The whole space forms a single orbit in the sense that given any two
points in the space there exists C G G transforming one point into the other.
In other words G acts transitively on X.
E x a m p l e 2.4. 2. Le t A
1
be a Euclidean n-space, and let G be the group of scale changes, that is, j? a G G,
E x a m p l e 2.4.3. Let X = X Y x X 2 where Xi is the p-dimensional Euclideanspace and X 2 is the set of all p x p symmetric positive definite matrices. Let
XeX uS€X 2 . Write
where X [t) is d; x 1 and 5 { i i ) is a\ x d; such that £ i U < * i = P- Let G be
the group of all nonsingular lower triangular pxp matrices in the block form,
S G G
9 a(X) = (aX 1 ,...,aX n )', a > 0.
X - ( X 1 i ) , . . . , X ( t ) ) ' ,
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Invariance 29
9 =
(9(H) 0
3(21) 3(22)
0 >
0
operating as
Let
where
\9(ki) 9(k2) ••• 9{kk)/
(X,S)->(gX,gSg').
f{X,S) = {Rl...,Rl),
£ ^ - % S ® % * = 1 . - . * (2-17)
Obviously, R* > 0. Let us show that Rj,..., if £ is a maximal invariant on X
under G. We shall prove it for the case k —2, The general case can be proved
in a similar fashion.
Let us first observe the following:
(i ) I f X -* gX, S - » <?5g' then - * 9 ( n ) X ( i ) , 5(u) - t 9 ( i i )S (n)0( U ) .
Thus ( i f j . f i f + R%) is invariant under G.
(ii) A " ' 5 _ 1 X - X J ' J J S J " ! , ^ ! ) + ( X ( 2 ) - 5(ai)5(n)^"(i))'
( 5 ( 2 2 ) - S ( 2 1 ) 5 ( ^ ) 5 ( i 2 ) ) " 1 ( X ( 2 ) - S(21)S l̂ \ ) X( 1) ) .
Now, suppose that for X,Y e X\\ S,T G A*2
( x ( ' 1 1 5 - ! ) x ( 1 ) , x ' 5 - 1 x ) - ( r ( ' 1 1 T - ; l y [ 1 ) , y ' r - 1 y ) , (2.18)
where K, T are similarly partitioned as X, S. We must now show that there
exists a.g eG such that X = gY, S = gTg'. Let us first choose
g = '0
with
3(21) 3(22)
3(11) =
3(22) ~ (S(22) ~ 5 ( 2 1 ) S ( ^ ) ) S ( 1 2 ) ) " " 2
9(12) = 9(22)5'(21)S (" 11 ) .
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30 Group Invariance in Statistical Inference
Th e n gSg' — I. Similarly choose
h = ( h { 1 1 ) 0
\ f y 2 l ) /l(22)
such that hTh! = I. Since
implies
we conclude that there exists an orthogonal matrix 0\ or order <ii x di suchthat
Since
X ' S - ' X = Y'T~l
Y,
s - 1
= s's,
T- 1 = h%
(fl(ii)-^(i))'(»(ii)^(i)) = (fc(u)%))'(tyii)%))>
then
I|S(21)-^(1) +9(22 |AT (2 ) | |2 = 11/1(211^(1) + ^(22)^(2) IP
so that there exists an orthogonal matrix 0 2 of order d 2 x d 2 such that
(9(21)X(i) + 9 ( 2 2 ) X ( 2 ) ) = 02( /l ( 2i)y"(i) +/l(22)V(2)) .
Now let
=f 0, 0 \
I o o 2 Jr -
Then, obviously,
X = g^ThY
= aY,
where a £ G and
gSg' = rhTh'V,
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Invariance 31
so that
S = aTo'
Hence (R J . f l J + i i j ) and equivalently (J?J, Ji^) * s a maxima! invariant on x
under G.
Note that if G is the group of p x p nonsingular upper triangular matrices
in the block form, that is, if g € G
9 =
( 9{ii) S(i2) ' • ' 9(i*) \
0 9(22) • ' • 9(2*)
V 0 0 •• • 9{kk)/
where is di x d{, then a maximal invariant on X under G is
( i r , . . . , n )
defined by
/ % ) • • • % , ]- l
Y^T* = (Xti ) ,,..,X m }' (X(i),... ,X w )i = 1, . . . ,k
2 .5 . D i s t r i b u t i o n o f M a x i m a l I n v a r i a n t
Let (X,A,X) be a measure space and suppose pg,9 £ 0, is a probability
density function w ith respect to A. Co nsider a group G operating from the left
on X and suppose that A is a left invariant measure,
\{A) = X(gA), g £ G, A £ A.
Let [yX) be a measurable space and suppose that T{X) is a measurable
mapping such that T is a maximal invariant on X under G. Let A* be a
measure on {y,C) indu ced by A, such that
X'(C) = X(T-1(C)), CeC
Also let Pi be the probability measure on (y,C) defined by
Pt{C) = j Pe(x)dX(x)!
JT-i(c)
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32 Group /n variance in Statistical Inference
so that P 2 has a density function p" with respect A*. Hence
/ p'{T)d\'{T) = j p(x)dX{x),C€C
A s s u m p t i o n . There exists an invariant probability measure p on the
group G under G.
L e m m a 2.5.1. The density function p* of the maximal invariant T with
respect to A* is given by
P*(T) = J P(gx)dp(g). (2.19)
Proof . Le t / be any measurable function on (y,C}- Then
j f(T)p'(T)d\-(T) = J f(T(x))p(x)d\'x).
However,
j f(T(x)) J p{g(x))dft(g)dX(x)
= j j HT(x))p(g(x))d\(x)du(g)
= J J nnx^pigix^dHgix^g)
= j f(T{y)) P{y)d\{y).
Then , since the function
/ p(g(x))dfi{g) JG
is invariant, we have
p*{T) = / p(g(x))M9) • Ja
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Invariance 33
2.5.1. Existence of an invaraint probability measure on O(p)
(group of p x p orthogonal matrices).
Let X = ( X i , . . . , X P ) be a p x p random matrix where the column vec
tors Xi (p-vectors) are independent and are identically distributed nor mal
(p-vectors) with mean vector 0 and covarian ce matrix / (identity). Ap ply theGram-Schmidt orthogonalization process to the columns of X to obtain the
random orthogonal matrix Y = (¥%,...., Y p ), where
Yi- g , i = i t . . . , p .
Define Y = T(X) and let G = 0(p). Obviously,
gY = (gY u ...,gY v )
= T(gX u ...,gX p )
= T(X;,...,X;),
where
X;=gXi, i = l , . . . . P .
Since X ; , X " have the same distribution and Xi,...,X* are independently
normally distributed with mean 0 and covariance matrix / , we obtain Y and
gY have the same distribution for each g £ 0(p). Thus the probability measure
of Y defined by
P[Y eC) = P(X eT~ L(C))
is invariant under 0{p),
2 . 6 . A p p l i c a t i o n s
E x a m p l e 2.6.1. ( N o n - c e n t r a l W i s h a r t d i s t r i b u t i o n ) . L e t X =
( X ! , . . . , X n ) (each X* is a p-vector) be a p x n matrix (n > p) where each
column Xi is norm ally distributed with mean vector / i ; and covariance matr ix
/ and is independent of the remaining columns of X . T h e probability density
function of X with respect to the pn-dimensional Lebesgue measure is
i&'t$ = ( 2 n - ) n p / 2 e x p { ~ i t t x x ' - | * i W * ' + t *a v i' j (2-19)
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34 Group Invariance in Statistical In/erence
where
fi = (ft , . , . . . , Up).
It is well known that
n
iml
is a maximal invariant under the transformation
X - XT, r e 0{n),
where O (n ) is the group of orthogonal matrices of dimension n x n . Le t p(s, p.)
be the probability density function of 5 at the parameter point p. with respect to the induced measure dX'{S) as defined in Lemma 2.5.1. T o find the
probability density function of 5 with respect to d we first find dX'{s). From
(2.13)
p(s,0 )dX'(s) = C(det s ) ( " - p - , ) / 2 e x p | - | t r s J da, (2.20)
By (2.18)
P(s,o)= f P ( 2 r r , o ) d ^ ( r )
Join)
- fH/ 0„.,d
*( r )
r e x p i - ^ t r s l ,(2ff)»f/ a P \ 2
where /i is the invariant probability measure on O( n ) . H ence,
dA*(s) = (2T) n " / 2 C ( d e t »f*+-W*it .
Again by (2.18)
p{a,u) = exp | -1 tr (s 4- J V ) J jf ^ exp j - j t r acT**' J dp{T)
= ^ P | - g tr(s+ W 0 j p ) ,
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Invariance 35
where
h(x,fi)= j exp J — - t r a d ? a ' \ du(T) Jot,n) I 2 J
It is evident tha t
h(x,u) = h(xT,p.<B),
where T, * e 0 {n). Thus we can write h(x,p) as
The probability density function of S with respect to the Lebesgue measure
ds is given by
p( s ,^)dA*(s ) =p( s ,^) (27r ) n p / 2 C(det s ) ( " - ' ' - 1 > / 2 d s .
This is called the probability density function of the noncentral Wishart dis
tribution with non-centrality parameter fia'.
E x a m p l e 2 . 6 .2 . ( N o n - c e n t r a l d i s t r i b u t i o n o f t h e c h a r a c t e r i s t i c
roots) . Let S of dimension p x p be distributed as central Wishart random
variable with parameter £ and degrees of freedom n > p. Let Ri > R 2 >
••• > R p > 0 be the roots of characteristic equation de t(S — XI) — 0, and
let P denote a diagonal matrix with diagonal elements R i , . . . ,R P . Denote by
p(i?, S) the probability density function of R at the parameter point E with
respect to induce measure dX'(R) of Lemma 2.5.1. We know [see, for example,
Giri(1977)]
/ p v ( n - p - l ) / 2 . p .
p(R,I)dX-(R) = C 1 {U^J e x p | 4 X > j *ll(&-Rj)dR
where dR stands for the Lebesgue measure and C\ is the universal constant.
It may be checked that R is a ma xima l invariant under the transformation
s r s r , r e o (p).
By (2.18)
p(R, I) = C 2 (j[ R^j exp | - \ J £ Ri | ^ dp(T).
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36 Group Invariance in Statistical Inference
So
dX'(R) = ^-'[[(R1-R ] )dR.
Again by (2.18)
/ p \ ( n - p - l ) / 2
p ( R , E ) = (det S " 1 ) " / 2 (n )̂
( n - p - l ) / 2
/ exp J --tr 0 -l
TRr'\ da(T) Jotv) I 2 J
x'O(p)
where t9 is a diagonal matrix whose diagonal elements are the characterist ic
roots of £ . The non-central probability density function of R depends on £
only through its characteristic roots.
2 .7. T h e Di s t r i bu t i on of a M a x i m a l Invar iant in the
G e n e r a l C a s e
Invariant tests depend on the observations only through the maximal invari
ant. To find the optimum test we need to find the explicit form of the maximal
invariant statist ic and its distribut ion. For many multivariate testing problems
it is not always convenient to find the explicit form of the m axi ma l invaraint .
Stein (1956) gave the following representation of the ratio of probability den
sities of a maximal invariant with respect to a group G of transformations g
leaving the testing problem invariant.
T h e o r e m 2.7.1. Let G be a group operating from the left on a topological
space (X, A) and \ a measure in x which is left invariant under G (G is
not necessarily transitive on X). Assume that there are two given probability
densities pi,p 2 with respect to A such that
Pi(A) = J AP2(x )d\( x ),
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Invariance 37
for A € A and Pi and P% are absolutely continuous. Suppose T : X —* X is a
where p, is a left invariant Haar measure on G. An often used alternative form
of (2.21) is given by
where f is the probability density with respect to a relatively invariant measure
with multiplier X(g).
Stein (1956) gave the statements of Th eo re m 2.7.1. without stating explic
itly the conditions under which (2.21) holds. However this representation was
used by Giri (1961, 1964, 1965, 1971) and Schwartz (1967). Schwartz (1967)
also gave a set of conditions (ra ther complicated) which must be satisfied for
Stein's representation to be valid. Wijsman (1967, 1969) gave a sufficient con
dition for (2.21) using the concept of Cartan G-space. K oe hn (1970) gave
a generalization of the results of Wijsman (1967). Bonde r (1976) gave some
conditions for (2.21) through topological arguments. Anderson (1982) has ob
tained some results for the validity of (2.21) in terms of the proper action of the
group. Wijsman (1985) studied the properness of several groups of transfor
mations used in several multivariate testing problems. Subsequently Wijsman
(1986) investigated the global cross-section technique for factorization of mea
sures and applied it to the representation of the ratio of densities of a maximal
invariant.
2. 8. A n I m p o r t a n t M u l t i v a r i a t e D i s t r i b u t i o n
Let Xi,..., XN be N independently, identically distributed p-dimensional
normal random variables with mean £ = ( £ , . . . , £p) ' and covariance matrix £
(positive definite). Write
maximat invariant and p" is the distribution of T(X} when X has distribution
Pi. Then under certain conditions
(2.21)
(2.21a)
N N
x =
w Ex
' - s = jyi x i -x%x i -xf.
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38 Group Invariance in Statistical Inference
It is well-known that X,S are independent in distribution and X has
p-dimensional normal distribution with mean £ and covariance matrix (1/jV)
E and S has central Wishart distribution with parameter E and degrees of
freedom n — N — 1. Throughout this section we will assume that N > p so
that S is positive definite almost everywhere. Write for any p-vector b,
where fofl is the d; x 1 subvector of b such that £j d* — p and for any p x p
matrix A
A ( n ) ••• A ( l f c ) N
, A ( A i ) • • • A(kk) j
Am =
' A n ) ••• A
(n)
where is a di x dj submatrix of A. Let us define Ri,... ,R k and ft,...,5*
by
t
i
E ^ = ^w2r7,̂ W, t = l,2,...,fc.
T h e o r e m 2.8.1. TTte joint probability density function of R% t . ..,R k is
given by (for Ri > 0, £ * = i fl^ < y
m « w - r ( w / 2 )
* / k \ [ ( W - P ) / 2 ] - l
x T j £ ( * / 2 W l - E ^ J
n / J V - <5,_i dj JL& \
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Invariance 39
where a — S i = i dj w ^ a o —0 and tp(a, b : x) is the confluent kypergeometric
series given by
a a{a + 1 ) x 2 a(a + l )(a + 2) x 3
+ b X + b(b + 1) 2! + 6(6 + l)(b + 2) 3!" + " ' '
Furthermore, the marg ina l probability density function of Ri,...,Rj,j < k
can be obtained from (2.22) by replacing A; by j and p by Y J[=I hi
proof . We will first prove the theorem for the case k — 2 and then use
this result for the case k — 3. T h e proof for the general case will follow from
these cases. Fo r the case k = 2 consider the random variables
W = 5(22) - 5( 21 )5 ^, 5( 12 ) ,
f = % , ) .
It is well-known [see, for example, Stein (1956), or Giri( l977) , p. 154] that
(i ) V is distributed as Wishart W(N - 1, E ( U ) ) ;
(ii) W is distributed as Wishart W{N - 1 - di, E [ 2 2 ) - S ^ i j S m j E ^ ) ) ;
(hi) the conditional distribution of L given V is multivariate normal with mean
£ ( 2 1 ) 2 ^ , V - 1 / 2 and covariance matrix
( £ ( 2 2 , - E ( 21) S ^ 1 i ) S ( 12 ) } ®/ ( ( 1 ,
where ® is the tensor product of two matrices £ (22 )— ^(21)^(11)^(12) and the
di x d, identity matrix I a
,•
(iv) W is independent of (L,V).
Let us now define W\, W 2 by
W 2(l + W L ) = N(X {2) - S m S ^ t i ) T ( S m - 5 ( 3 i ) 5 ( - 1 ) 5 ( i 2 ) ) - 1 (2.24)
x ( X ( 2 ) - 5 ( 2 i , 5 ^ , A " ( 1 ) ) .
Since X is independent of 5, the conditional distribution oi\fNX ̂ 2) given 5 ( n j
and X(i) is normal with mean \/~N {£ (2) + ^<2i )^( i i )£ ( i ) ) and covariance matrix
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40 Group Invariance in Statistical Inference
£{22> ~ E{8 i>Er i i jE ( i 4> 1 1 thereupon follows that the conditional distribution
of y/NXt2) given 5 ( l l , and y/NX^f is independent of the conditional distri
bution of
given Sni) and Xay. Further more, the conditional distribution of VWS(i2)
5(~!)^{ i ) give 1 1 s ( n ) a n ( i i s normal with mean V W E ( a i ) S ^ j j ^ i ) and
covariance matrix
Hence the conditional distribution of
Viv" ( x < 2 ) - ( i + W M _ 1 / 2
given 5( H ) a n d -^(i) ' s normal with mean
(C(2) - S ( 2 i ) S ^ i ^ ( i ) ) ( i + Wi)-1? 2
and covariance matrix
£ ( 22 ) - ^(21) £ ( i i ) £ ( 1 2 ) -
From Giri (1977) it follows that the conditional distribution of W% given X^
and S(u) is
p(W 2 \X (1) ,S tn) ) = xl [Hi + W i ) " 1 ) /xl-dl-dl , (2.25)
where Xn($) denotes the noncentral chisquare random variable with noncen-
trality parameter 6 and with n degrees of freedom and Xn denotes the central
chisquare random variable with n degrees of fredom. Since this conditional
distribution depends on Xh-t and Sti\) only through Wi, we get the joint
probability density of WuWz as
p{W 2
,Wi) =X
2
2
(6 2
(l + W i ) -
l
) / x N
.d l
. ^ (2.26)
and furthermore it is well known that the marginal probability density function
of Wi is
P(Wi) = X 2 dl (Si)/x N -ai • (2.27)
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Invariance 41
Hence, we have
= 0 jSI f l + i f f t ^ ^ H *
r(* + /?)r(V) 1 2 JJ3 j !
From Lemma 2.8.1. below,
W 2 = {(Ri + R2 ){1 - R i - R a ) " 1
~H i (l- Ri)~1 }
x (1 + Hj.{l - H i ) " 1 ) " 1
= fi2(l-fli-fi2)"1.
(2.28)
(2.29)
From (2.28) and (2.29) the probability density function of Ri and R 2 is
given by
x R$ 4I -1 BI*'-1(I-RT - nriW-Jh*
x exp | - - ( f t +6 2) + rfefiji
2
x n ^ i ( J V - f f l _ 0 , f ; i i U , ) (2.30)
We now consider the case k = 3. Write
W 3 - ( N X ' S "1
: ? - N%S^X{ 3 y) ( l +
It is easy to conclude that
• 1
5(33, - 5[32jS [ 2 2 ] S l 2 3 j
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42 Group /n variance in Statistical Inference
is distributed as Wishart
w(N-l-d-!- da,E(33) - E[33]E^jE(23]j ,
and is independent of S( 22) and S[ S2-• Furthermore, the conditional distribution
of \ / iVX(3) given S[ 22 ]<X [2] is normal with mean
(fo) - E|32]S|2 2 J(-?[2] - £[2]))
and covariance matrix
E(33) - E[32]Ep2] S[23] .
and is independent of the conditional distribution of
^S
[M ] S
\ ~22) X
W
given S[Z2] and X p ] , which is normal with mean
and covariance matrix
NX
m
S
[~2
l
2)
X
m (
S
[33] -
S
(32]
S
[22j
S
[23]) -Thus the conditional distribution of W 3 given Sp2j and X[ 2 j or, equivalently
given (Wi,W 2 ), is
viW^W^Wy) = X %$41 + WiftWi + W 2 + WiWsT 1
)
Ixfu-dy-it-di- (2-31)
Hence the joint probability density function of , W 2 , W 3 is given by
p(Wi,W 2 ,W 3 ) - — ^ 5 '-
XN-di-di-dz
X.N-d,-d? Aiv-fl,
From Le mma 2.8.1. below,
W 1=R1 (1 -Rl )-
1 ,
W a = R i ( l - R l - R 2 ) -1
,
W 3 = {{R, +R 2 + R 3 )(l - R , - R 2 - ^ p 1 - (R, + fl2)
x (1 - R, - R^-'Xl - R, - R 2 )
= R 3 ( . 1 - R l - R 3 - R a ) - 1 . (2.33)
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Invariance 43
From (2.32)-(2.33) the joint probability density function of i f , , i f 2 , # 3 is
given by
i=l V i=l
x n ^ i t i V - ^ f ; ^ ) . (2.34)
Proceeding exactly in the above fashion we get (2.22) for general k. Since
the marginal distribution of JEui is normal with mean &a and covariance matrix
S[iij it follows that given the joint probability density function of i f i f ^ ,
the marginal distribution of i f , , . . . , i f j , 1 < j < p can be obtained from (2.22)
by replacing k by j . •
The following lemma will show that there is one-to-one correspondence
between ( f i i , . . . , i ft ) and (if J , . . . , if£) as defined earlier in this chapter.
L e m m a 2.8.1.
NX'S^X NX'(S + NXX')-1 X =
1 + NX'S-i-X
Proof. Let
{S + NXX 1
)-1 = S - ' +A.
Hence,
I + NXX'S-1 +(S + NXX')A = I.
So we get
(S + NXX')-1 = S " 1 - N(S + NXX'^XX'S'1
Now
NX'(S + NXX')-^ = NX'S^X - NX'(S + NXXT'XiX'S^X).
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44 Group Invariance in Statistical Inference
Therefore,
NX' S~1 X NX><a+ NXXT*X)=1+NX , s_lx
Note that
5 = i j=i V J = I
2 .9 . Almost Invar iance , Suff ic i ency and In va r i an ce
Def in i t ion 2.9.1. Let (A" ,A ,P ) be a measure space. A test ih{X),X G
A" is said to be equivalent to an invariant test with respect to the group of
transformations G, if there exists an invariant test ip(x) with respect to G
such that
ib(x) = y{x),
for all x G X except possibly for a subset TV of A" of probability measure zero.
Def in i t ion 2 .9 .2 . Al mo st inva r ian ce . A test ip{x), x G X, is said to be
almost invariant with respect to the group of transformations G, if
<p(x) = tp(gx) for all x G X — J\f g ,
where N g is a subset of x depending on g of probability measure zero.
It is tempting to conjecture that if tp(x) is almost invariant then it is equiv
alent to an invariant test i>{x). Obviously, any ifi(x) which is equivaleant to an
invariant test is almost invariant. For, take N g — N U g-1 N. If x £ N g , then
x & N, and gx $ N so that
tp(x) = y(gx) = ih(gx) = ii>{x).
Conversely, if G is countable or finite then given an almost invariant test <p(x)
we define
N = 1J N g
,
where N g = {x G x '• f{x) # f{gx)} and AT has probability measure zero.
The uncountable G presents difficulties. Such examples are not rare where
an almost invariant test can be different from an invariant test.
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Invariance 45
Let G be a group of transformations operating on X and let, A,B, be the
d-field of subsets of X and G respectively such that for any A £ A the set of
pairs (x, g) for which gx £ A is measurable Ax B. Suppose further that there
exists a tr-finite measure v on G such that for all B £ B,v(B) = 0 implies
v{Bg) = 0 for all g £ G . Then any measurable almost invariant test functionwith respect to G is equivalent to an invariant test under G.
For a proof of this fact the reader is referred to Lehmann (1959). The
requirement that for all g £ G and B £ B,v(B) — 0 imply v(Bg) = 0 is
satisfied in par tic ula r when f{B) — f{Bg) for all g £ G and B £ B. Such a
right invariant measure exists for a large number of groups.
Let A\ = {A : A £ A and gA —A for all g £ G } and let .4 S be the sufficient
o--field of X. Le t G be a group, leaving a testing problem invariant , be suchthat gA„ = A s for each g £ G . If we first reduce the problem by sufficiency,
getting the measure space (X, A s , P ) , and subsequently by invariance then we
arrive at the measure space (X,A si, P) where A si is the c lass of all invariant
A s measurable sets. T he following theorem which is essentially due to Ste in
and Cox provides an answer to our questions under certain conditions. T h e
proof is given by Hall , Wijsman and Ghosh (1965).
T h e o r e m 2.9.1. Let [X, A, P ) be a measure space and let G be a group of
measurable transformations leaving P invariant. Let A s be a sufficient subfield
for P on (X, A) such that gA s — A s for each g £ G . Suppose that for each
almost invariant A 3 measurable function <p — tb except possibly on a set N of
probability measure zero. Then A s is sufficient for P on (X,A).
Thus under conditions of the theorem if we reduce by sufficiency and in
variance then the order in which it is done is immateria l.
2 .1 0. I n v a r i a n c e , T y p e D a n d E R e g i o n s .
The notion of a type D or type E region is due to Issacson (1951). Kiefer
(1958) showed that the P- te st of the univariate general linear hypothesis pos
sesses this property. Suppose, for a parametric space ft — {(#,i?) : 6 £ ft',n €
H] with associated distributions, with ft' a Euclidean set, that every test func
tion <fr has a power function 0${$,n} which, for each n, is twice continuouslydifferentiable in the components of 0 at 0 = 0, an interior point of ft'. Let Q a
be the class of locally strictly unbiased level a tests of Ho : $ — 0 against the
alternatives H\:0^O. Our assumption on implies that all tests in Q a are
similar and that
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46 Group Invariance in Statistical Inference
for all 4> in Q a .
Let be the matrix of second derivates of 0$(9, v) with respect to the
components of $ (which is also popularly known as the Gaussian curvature ) at6 = 0. Define
AM = d e t ( / V " ) ) -
Assumption. A^ (T J ) > 0 for all v E H and for at least one d>' 6 Q a .
D e f i n i t i o n 2 .10 .1. ( T y p e E an d T y p e D Te s t s ) . A test 4>' in Q a is
said to be of type E if
A ^ ( n ) - m a x ^ e Q = A ^ ( n )
for all V € H. If i f is a single point, d>~ is said to be of type D.
Lehmann (1959a) showed that, in finding regions which are of type D,
invariance could be invoked in the manner of Hu nt -S te in theorem. (Theorem
5.OA); and that this could also be done for type E regions (if they exist)
provided that one works with a group which operates on H as the identity.
Suppose that our problem is invariant under a group of transformations for
G which Hunt-Stein theorem holds and which acts trivially on fi', such that
for g e G
If (fig is the test function defined by
tpg(x) = <j>(gx),
then a trivial computation shows that
A * 9 ( n ) = A ^ o n )
and hence
A ( n ) = A{gr))
where
A(n) = m a x ^ ^ A ^ n ) .
Also, if 0 is better than 4>' in the sense of either type D or type E, then tpg is
clearly better than <p'g.
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Invariance 47
E x e r c i s e s
1. Let Xi,..., X n be independently and identically distributed normal random
variables with mean a and variance (unknown) I T 2
. Using Theorem 2.7.1.
find the U M P invariant test of Ho : p, = 0 against the alternative Hi : u ^ 0
with respect to the group of transformations which transform each Xi —*
cXi,c±Q.
2. Let {Pg,8 e 11} be a family of distributions on (X,A) such that P$ has a
density p(-j8) with respect to some a-finite measure u. For testing Ho • 0 €
flrj against Hj : 9 G Hi with 120
n P i = 0 ( n u ' l s e t ) . t n e likelihood ratio test
rejects Ho whenever X(x) is less than a constant, where,
sup p(x\9)
X(x) = 9€.n0
sup p(x|S)
9 e flj u Hi
Let the problem of testing r/o against i i ] be invariant under a group G of
transformations g transforming x —* g( x), g 6 G, i G A". Assume that
p(x\9)=p(9x\g8)X(9)
for some multiplier X. Show that A( x) is an invariant function.
3. Prove (2.8) and (2.25).
4. Prove Theorem 2.9.1.
References
1. S. A, Anderson, Distribution of maximal invariants
usingquotient
measures,Ann. Statist. 10, pp. 955-961 (1982).
2. D. Blacltwell, On a class of probability spaces, Proc. Berkeley Symp. Math.
Statist. Probability. 3rd, Univ. of California Press, Berkeley, California, 1956.
3. J. V. Bonder, Borel cross-sections and maximal invariants, Ann . Statist. 4,
pp. 866-877 (1976).
4. T. Ferguson, Mathematical Statisties, Academic Press, 1969.
5. N. Giri, On the likelihood ratio test of a normal multivariate testing problems;
Ph.D. Thesis, Stanford University, California, 1961.
6. N. Giri, On the likelihood ratio test of a normal multivariate testing problems,Ann. Math. Statist. 35, pp. 181-189 (1964).
7. N. Giri, On the likelihood ratio test of a normal multivariate testing problem-II,
Ann. Math. Statist. 36, pp. 1061-1065 (1965).
8. N. Giri, On the distribution of a multivariate statistic, Sankhya, 33, pp. 207-210
(1971).
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48 Group Invariance in Statistical Inference
9. N. Giri, Multivariate Statistical Inference, Academic Press, N.Y., 1977.
10. S, L , Isaacson, On the theory of unbiased tests of simple statistical hypotheses
specifying the values of two or more parameters, Ann, Math. Statist. 22, pp. 217¬
234 (1951).
11. A. Haar, Der Massbegriff in der theorie der kontinvjierchen gruppen, Annals of Mathematics, 34 (1933), pp. 147-169.
12. W. J . Hail, R. A. Wijsman, and J . K. Ghosh, The relationship between sufficiency
and invariance with application in sequential analysis, Ann. Math. Statist. 36,
pp. 575-614 (1965).
13. P. R. Halmos, Finite Dimensional Vector Space, D. Van Nostrand Company,
Inc. Princeton, 1958.
14. J . Kiefer, On the nonrandomized optimatity and randomized nonoptimality of
symmetrical designs, Ann. Math. Statist. 29, pp. 675-699 (1958).
15. U. Koehn, Global cross sections and the densities of maximal invariants, Ann.
Math. Statist. 41, pp. 2045-2056 (1970).
16. E . L . Lehmann, Testing of Statistical Hypotheseses, John Wiley, N.Y., 1959.
17. E . L. Lehmann, Optimum invariant tests, Ann. Math. Statist. 30, pp. 881-884
(1959a).
18. L, Loomis, An Introduction to Abstract Hermonic Analysis, D. Van Nostrand
Company, Inc., Princeton, 1948.
19. L. Nachbin, The Haar Integral., D, Van Nostrand Company, Inc., Princeton,
1965.
20. R. Schwartz, Locally mimimax tests, Ann. Math. Statist., 38, pp 340-359, 1967.
21. C . Stein, Multivariate Analysis-I, Technical Report 42, Department of Statistics,
Standford University, 1956.
22. R. A. Wijsman, Cross-Section of orbits and their application to densities of max
imal invariants, Proc. Fifth. Berk. Symp. Math. Statist. Prob. 1. pp. 389-400
(1967) University of California, Berkeley.
23. R. A. Wijsman, General proof of termination with probability one of invariant
sequential probability ratio test based on multivariate observation, Ann. Math.
Statist. 38, pp. 8-24 (1969).
24. R. A. Wijsman, Distribution of maximal invariants, using global and local cross-
sections, Preprint, Univ. of Illinois at Urbana-Champaign, 1978.
25. R. A. Wijsman, Proper action in steps, with application to density ratios of max
imal invariants, Ann. Statist. 13, pp. 395-402 (1985).
26. R. A. Wijsman, Global cross sections as a tool for factorization of measures and
distribution of maximal invariants, Sankhya A, 48, pp. 1-42 (1986).
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Chapter 3
EQUIVARIANT ESTIMATION INCURVED MODELS
Let ( x , A ) be a measure space and f) be the set of points 8. Denote by
{Ps,8 £ Si], the set of probability distributions on (x, A ) . Let G be a group
of transformations operating from the left on \ such that p £ G , g ; x —* X
is one to one onto (bijective) . Le t G be the corresponding group of induced
transformations g on ft.
Assume
(a) For 0i e 0, i = 1,2; 9i ? t9 2 ,P e , £ P B „
(b) P f l { A ) - P- ge(gA), A £ A,p € G,g e G.
Let A(0) be a max im al invariant on ft under G , and let
f!" = {0\9 £ fi.with A(<?) = A 0 } (3.1)
where Ao is known. We assume x to be the space of minimal sufficient statis tic
for 8.
Defini t ion 3 .1 . ( E q u i v a r i a n t Estimator) A point estimator 8(X),X £
X, a mapping from x —' ft, is equivariant if 8(gX) = g8(X),g £ G .
Note 1. For notational convenience we are taking G to the group of transformations on
e(X).
Note 2. The unique max imum likelihood estimator is equivariant.
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50 Group Invariance in Statistical Inference
L et T(X),X £ X' be a maximal invariant on x under G. Since the distri
bution of T(X) depends on 0 € Q only through A(0) , given X($) = A 0 T ( X )
is an anc illary statistic . A n anc illary stat ist ic is defined to be a part of the
minimal sufficient statistic X whose marginal distribution is parameter free.
In this chapter we consider the approach of finding the best equivariant es
timator in models admitting an anci llary statis tic. Such models are assumed
to be generated as an orbit under the induced group G on fi. Th e ancillary
statistic is realized as the maximal invariant on x under G. A model which
admits an ancil lary statist ic is referred to as a curved model. Models of this
nature are not uncommon in statistics. Fisher (1925) considered the problem
of estimating the mean of a Normal population with variance a 2 when the
coefficient of variation is known. The motivation behind this was based on theempirically observed fact that a standard deviation a often becomes large pro
portionally to a corresponding mean p so that the coefficient of variation a/p
remains constant. This fact is often present in mutually correlated multivariate
data. In the case of multivariate data , no well-accepted measure of variat ion
between a mean vector p and a covariance matrix E is available. Kariya , Giri
and Perron(1988) suggested the following multivariate version of the coefficent
of variation, (i) A = fl'SS-1
/! (ii) v = S " 1 ^ where E = E ^ E 1 ' 2 with S 1 / 2
as the unique lower triangular matrix with positive diagonal elements.
In recent years Cox and Hinkley (1977), Efron (1978), Amari (1982 a,b)
among other have reconsidered the problem of estimating p, when the coefficent
of variation is known in the context of curved models.
3 .1 . Best E q u i v a r i a n t E s t i m a t i o n o f fi wi t h A K n o w n
Let Xi,...,X n (n > p) be independently and identically distr ibuted JVp
(p, E ) . We want to estimate a under the loss function
L{n,d) = (d-n)'-L-1{d-p) (3.2)
when A = a'T,~l p, — A 0 (known). Let
n 7i
nx = Y ,x>,s = - x)(Xi -xy.
Then (X,S) is minimal sufficient for {u, E ) , JUX is dis tributed independently
of S as N p(y/nu, S ) and S is distributed as Wishart Wp(n - 1, E ) with n — 1
degrees of freedom and parameter E . Under the loss function (3.2) this problem
remains invariant under the group of transformations Gj(p) of pxp nonsingular
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Equivariant Estimation in Curved Models 51
matrices g transforming (X,S) —» (gX ,gSg'). The corresponding group G of
induced transformations g on the parametric space ft transforms 8 = (p., £) —*
g8 — (pp, o £ p ' ) . A maximal invariant invariant on the space of (\/nX, S) is
T 2
= n(n- l J X ' S - ' X (3.3)
and the corresponding maximal invariant on the parametric space ft is A =
(i'T,~1u. Since the distribution of T 2 depends on the parameters only through
A, given A — A o T 2 is an ancillary statistic . Since the loss function L[p,d) is
invariant under the group Gi(p) of transformations g transforming {X,S) —»
{gX,gSg') and ( « , S ) —• (gp,gT,g') we get for any equivariant estimator p. of
R(8,( l) = Ee(p-p)'^- ,(p-p,))
= Ee(gjl - ga)'{g , £,g')~ 1{gii. - gp))
= Ee(p(gX) - 9${gWT\mm - gp)
= Ege(a{X) - g n)'{gZ^y\^{gX) - gp)
= * ( 5 M ) , (3-4)
for g € Gt{p),g is the induced transformation on E corresponding to g on x-
Since G|(p) acts transitively on E we conclude from (3.4) that the risk R(8,fi)
of any equivariant estimator ji is a constant for all 8 € ft.
Taking A 0 — 1, which we can do without any loss of generality and using
the fact that R(8, ii) is a constant we can choose p..= e = ( 1,0, . • . , Q) , E = I.
To find the best equivariant estimator which minimizes R(8,p) among all
equivariant estimators p. satisfying
ji(gX,gSg')=gp(X,S)
we need to characterize p.. Let Gr{p) be the subgroup of G;(p( containing all
p x p lower triangular matrices with positive diagonal elements. Since 5 is
positive definite with probability one we can write, S — W W ' , W £ Grip)-
Let
v = w~lY
'Q = WT\
where Y = •JnX, \\ V \\ 2= V'V = T 2 /(n - 1).
T h e o r e m 3.1. If p, is an equivariant estimator of p, under Gi(p) then
p(Y,S) = k(U)WQ (3.5)
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52 Group Invariance in Statistical Inference
where k is a measurable ofU = T 2 j(n — 1).
Proof. Since p is equivariant under Gi (p ) we get for g € Oi(p)
gp(Y,S) = p.(gY,gSg'). (3.6)
Replace Y by W _1Y, g by W and S by I in (3.6) to obtain
p(Y,S) = Wp(V,I). (3.7)
Let O be an pxp orthogonal matrix with Q = V/ || V \\ as its first column.
Then fi(V,I) = p(00'V,00') = OH{VUeJ).
Since the columns of O except the first one are arbitrary as far as they are
orthogonal to Q it is easy to claim that the components of p.(\fUe,I) except
the first component p L(VU e, / } are zero. Hence
p{V,I) = Qji1{y/UeJ). •
The following theorem gives the best equivariant estimator (BEE) of p.
T h e o r e m 3.2 . Under the loss function (3.2) the unique BEE of p isp = k
(If) WQ where
k(U) = E(Q'W'e\U)/E(Q'W'WQ\U). (3.8)
P r o o f . B y Theo rem 3.1 the risk function of an equivariant estimator p,
given Ao — 1, is
R(6,p)^R((e1 I),,l)
= E(k(U)WQ - e)'(k(U)WQ - e).
Using the fact that U is ancillary, a unique BEE is obtained as p = k(u)WQ
where k(u) minimizes the conditional risk given U
E{((k(U)WQ - e)(k{U)WQ - e)')\U\.
This implies k(U) is given by (3.8). •
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Equivariant Estimation in Curved Models 53
3.1.1. Maximum likelihood estimators
The maximum likelihood estimators ft and £ of p and £ respectively under
the restriction An = I are obtained by maximizing
- | l o g (det £ ) - | tr 5 S "1
- | j £ - M J ' E - 1 ^ - M ) - (3.9)
with respect to p, £ where 7 is the Lagrange multiplier. Maximizing (3.9) we
obtain
p—~ —, S - - 5 + 7 M ' ( 7 + C I . (310)n + 7 71
Using the condition J t ' E - l / i = 1 we get
. (U-^U(4 + 5U)\ ^
(3.11)
The maximum likelihood estimator (mle) p is clearly equivariant and hence it
is dominated by jl of Theorem 3.2 for any p. In the univariate case the mle is
71 4(3.12)
Thus (3.12) is the natural extension of (3.11). Hinkley (1977) investigated
some properties of this model associated with the Fisher information. Amari
(1982 a, 6) proposed through a geometric approach what he called the dual mle
which is also equivariant.
3.2. A Particular Case
As a particular case of A constant we consider now the case £ = J when
C 2 is known. Let Y = y/nX,W = tr S. T h en (Y,W) is a sufficient statistic
for this problem, and i s distributed independently of Y as X ( n - i ) p -
We are interested here to estimate /t with respect to the loss function
(3.13)
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54 Group Invariance in Statistical Inference
This problem remains invariant with respect to the group G = R+ x 0(p),R+
being the multiplicative group of positive reals and O(p) being the multiplica
tive group of p x p orthogonal matrices transforming
Xi —t bTXi , t = 1 , . . . , n
d -4. bTd,
^gf) a ~ n ,
where (6,T) € G with b e R+,T g O(p). The transformation induced by G on
(Y, W) is given by
(Y,W) — (!>rY, f>2
W).
The theorem below gives the representation of an equivariant estimator
under G.
T h e o r e m 3.3. An estimator d(Y, W) is equivariant if and only if there
exists a measurable function h ; R+ —* R such that
d{Y,W) = h{^p}Y
for all [Y,W)£R? x R+ .
Proof . I f h is a measurable function from if+ —* R and d[Y, W) —
h (^w~j" Y t n e n clearly d is equivariant under G . Converse ly if d is equiv
ariant under G , then
(VY W\ d(Y,W) = b r d { — , ¥ ) (3.14)
for all T £ 0{p),Y € R p ,b > 0 , W > 0. We may assume without any loss of
generality that Y'Y > 0.
L e t Y, W be fixed and
- f t )where di is the first component of the p-vector d. Let A € O(p) be a fixed
p x p matrix such that
Y'A = (\\Y | | , 0 , . . . , 0 )
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Equivariant Estimation in Curved Models 55
Let A be partitioned as A = ( A J . A J ) where A, = \\Y\\~1Y. Now choose
T = {Ai,A 2 B) with B e 0(p-l) and b = \\Y\\. From (3.14) we get
d(Y,W) = d 1 ^ ( l , 0 . . . 0 ) . ^ ) r .
+ \\Y\\A 3 Bd a((l t0...Q) t~) . (3.15)
Since (3.15) holds for any choice of B € 0(p — 1) we must have then
d(Y,w) = d l ( ( i , o . . . o ) . ^ ) r . •
It may be easily verified that a maximal invariant under G in the space of
sufficient statistic V = W~l Y'Y and a corresponding maximal invariant in
the parametric space is
As the group acts transitively on the parametric space the risk function (using
(3.13))
fl(M) = £ „ ( £ ( M ) )
of any equivariant estimator d is constant. Hence we can take p = po =
[C, 0.. . 0)' . Thus the risk of any equivariant estimator d can be written as
fl(po,d) = (L (p0 ,k y ) )
= Eu0(E(L(p n ,h(^-} r ) ) V = *) •
To find a B E E we need to find the function ho satisfying
EpQ(L(p0 ,h0(V)Y\V = v)< Ep0{L{po,h{v)y)\V = v)
for all h : R+ —* R measurable functions and for all values v of V. Since
EpampoM^YW = v)
= h 2
(v)Epo(Y'Y\V =v)~ 2h(v)Epo(Y l \V = v) + l
where K = ( y l t . . . , y p ) \ we get
EMY i \V = v) f 3 . 6 1
k o { v ) - Ep0(Y'Y\V = v)- ( 3 ' 1 6 )
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56 Group Invariance in Statistical Inference
T h e o r e m 3.4. The BEE da(xlt .. .,X„;C) = d 0(Y,W) is given by
d 0{Y,W)
' ™ rfjny+i+l) (n& ^
nC 2 hr i^+i+l )v\ 2
V r t i " P + j + i ) / /
B Ct
V
h hir+W \ 2 J .
(3.17)
where t — y ( l + ti)—1
Proof . The joint probability density function of Y and W under the as
sumption fi — fi0 is
[ e x p { - ( C2
/ 2 ) ( j / y - 2 y 5 g i + n + I K ) } ^ ' " -1
' " -1
/ ( y > ) = { 2 " p / 2 ( C 2 ) " " p / 2 ( r ( i ) ) p r ( ( n - l )p /2 ) , if w > 0
0, otherwise.
Hence the joint probability density function of Y, V is given by
(• e x p{ - ( C 2 / 2 [ ( ( l + i Q W i , + w ] }
/ y „ ( „ „ j = 2 ^ 2 ( C 2 ) - ^ \ r ^ m ( n - l )p /2 )
I 0, otherwise.
Using (3.16) we get (3.17).
C o r o l l a r y . If m - p(n - l ) / 2 is an integer, the B E E is given by
n <" 2
do(Y,W)=T ^h0(v) = g(t)X,
with <j(t) = p(t)/w(t) where
m+l
u(t) = y ,
m + l
m+ 1
m + l \ / ' n C3
\ 3 ...
Idr ( | p + i )
(3.18)
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Equivariant Estimation in Curved Models 57
Proof. Let Y k be distributed as %%• Then
c . / y n ^ _ o t t r ( f c / 2 + a ) -kE
^ ~2
r(fc/2)l f Q >
^ " '
Hence with m as integer
t E i Y z ^ M - n C H m ^ y t
h0(v) = v ^ C 2 ^
£ ) e x p { - n C 2 ( / 2 } ( ^ ) J 43=0
= v
/ S C 2 £ ( V 1
m ) / £ ( V 2
m + 1 ) , (3.19)
where Vj is distributed as noncentral X 2 ,+2( n( ~' 2i)^ and ^2 is distributed as
noncentral Xp( nC 2 t). Hence letting V = ^ ( t f 2 ) a ° d taking r as an integer we
get
Prom (3.19) and (3.20) we get (3.18). •
Note 1. g(t) is a continuous function of f, and
lirn S (f) = T i C2
/ p ,(-.0+
hmg(t) = g { l ) < l ,
and g(t) > 0 for all t. Thus when Y'Y is large the B E E is less than X.
Note 2. We can also write d Q = (1 - ^ ) X where r(t))/t) = 1 - g(t). This
form is very popular in the literature. Perron and Giri (1990) have shown that
g{t) is a strictly decreasing function of t and T(V) is strictly increasing in v. T h e
result that g(t) is strictly decreasing in f tells what one may intuitively do if he
has an idea of the true value of C and observe many large values concentrated.
Normally one is suspicious of their effects on the sample mean and they havethe tendency to shrink the sample mean towards the origin. That is what our
estimator does. The result that T ( V) is strictly increasing in v relates the B B E
of the mean for C known with the class of minimax estimators of the mean for
C unknown. Efron and Morris (1977) have shown that a necessary condition
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58 Group Invariance in Statistical Inference
for an equivariant estimator of the form g{t)X to be minimax is g(t) —* 1 as
t —* 1, SO our estimator fails to be minimax if we do not know the value of
C. On the other hand Efron and Morris (1977) have shown that an estimator
of the form d = (1 — Z^p-)X is minimax if (i) r is an increasing function, (ii)
0 < T ( V ) <(p- 2 ) / (n - 1) + 2 for all v e (0, oo). Thus our estimator satisfies
(i) but fails to satisfy (ii). So a truncated version of our estimator could be a
compromise solution between the best, when one knows the values of C and
the worst, one can do by using an incorrect value of C.
3.2.1. An application
The following interesting application of this model is given by Kent, Briden,
and Mardia (1983). The natural remanent magnetization ( N R M ) in rocks is
known to have, in general, originated in one or more relatively short time inter
vals dur ing rock-forming or metamorphic events during which N R M is frozen in
by falling temperature, grain growth, etc. The N R M acquired dur ing each such
event is a single vector magnetization parallel to the then-prevailing geometric
field and is called a component of N R M . By thermal, alternating fields or chem
ical demagnetization in stages these components can be identified. Resistance
to these treatments is known as "stability of remanence". At each stage of
the demagnetization treatment one measures the remanent magnetization as a
vector in 3-dimensional space. These observations are represented by vectors
Xi,...,X n in fi3. They considered the model given by j j j — a , +0i+ei where
c»i denotes the true magnetization at the ith step, 0i represents the model er
ror, and ei represents the measurement error. They assumed that 0, and e<
are independent, p\ is distributed as N 3{0,T 2{c>i)I), and e z is distributed as
AM0>cr 2 (a,) / ) . T h e Q ; are assumed to possess some specific structures, like
collinearity etc., which one attemps to determine. Sometimes the magnitudeof model error is harder to ascertain and one reasonably assumes r 2(a) = 0.
In practice <J 2{a) is allowed to depend on a; and plausible model for c 2(a)
which fits many data reasonably well is cr 2(a) — a{a'o) + b with a > 0, b > 0.
When a'a large, b is essentially 0 and a is unknown.
3.2.2. Maximum likelihood estimator
The likelihood function of xi,... x n with C known is given by
L(xi,...,x n \p)
= C ^ ) - - / 2 e x p | - ^ - ( ( , + j / ' J , - 2 ^ ' M + V M ) } •
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Equivariant Estimation in Curved Models 59
Thus the mle p of p (if it exists) is given by
[{np/C 2 )(p'p) -w-y'y + 2 \ / n i / A ] £ = Vnp'py • (3.21)
If the E q . (3.21) in ft has a solution it must be colinear with y. Let p = ky be
a solution. From (3.21) we obtain
k[(np/C 2 )y'y)k 2 + Jn~y'yk - {y'y +w)] = 0.
Two nonzero solutions of A; are
- 1 - ( 1 + ^ ( ^ ) )1 / 2 - 1 + (1 + ^ ( ^ ) )
1 / 2
1 _
iJkpIC 2 ' 2 ~ 2^lp/C 2
To find the value of k which maximizes the likelihood we compute the
matrix of mixed derivatives
a 2 ( - logL)
dp'dp kHy'y)'
and assert that this matrix should be positive definite. T h e characteristic roots
of this matrix are given by
k 3 y'y 'A 2 -
x/nC 2 + 2npk
k 2 y'y
If k = ki, then Ai < 0 and A 2 < 0. B ut if k = k 2 , then A, > 0, A 2 > 0.
Hence the mle p. — d-i(xi,... ,x n ,C) is given by (corresponding to positive
characteristic roots)
d 1{x1 ,...x n ,C) =(1 + 4 p / C
2
( )1 / 2 - 1
2pC 2 X. (3.22)
Since the maximum likelihood estimator is equivariant and it differs from
the B E E do the mle d\ is inadmissible. The risk function of do depends on C .
Perron and Giri (1990) computed the relative efficiency of do when compared
with d]_, the James-Stein estimator (d 2 ), the positive part of the James-Stein
estimator (d 3
), and the sample mean X ( d j ) for different values of C, n, and p.They have concluded that when the sample size n increases for a given p and C
the relative efficiency of do when compared with di, i = 1,.. , 4 does not change
significantly. This phenomenon changes markedly when C varies. When C is
small, dp is markedly superior to others. On the other hand, when C is large
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60 Group Invariance in Statistical Inference
all five estimators are more or less similar. These conclusions are not exact as
the risk of d 0 ,di are evaluated by simulation. Nevertheless, i t give us sufficient
indication that for small values of C the use of BEE is clearly advantageous.
3 .3 . Bes t E q u i v a r i a n t E s t i m a t i o n i n C u r v e d C o v a r i a n c e M o d e l s
Let Xi,... ,X n(n > p > 2) be independently and identical ly distributed
p-dimensional normal vectors with mean vector p and positive definite conva-
riance matrix S.
Let E and S = (X a - X)(X a - Xf be partitioned as
/ i P-I V ( i p - i V
£ — 1 £ l l ^12 I . 5 — 1 Sn S\ 2 I
We are interested to find the BEE of 0 — E ^ ^ i on the basis of n sample
observations Xi,...,x„ when one knows the value of the multiple correlation
coefficent p 2 — E ^ E ^ sE ^ E21. If the value of p 2 is significant, one would
naturally be interested to estimate 0 for the prediction purpose and also to
estimate £22 to ascertain the variabili ty of the predict ion variables.
Let Gi(p) be the full linear group of p x p nonsingular matrices and let
0{p) be the multiplicative group of p x p orthogonal matrices. Let Hi be the
subgroup of Gi(p) defined by
/ ' P - 1
Hi = {ft € Gi(p): ft = A n 0
\ 0 h22
and let H 2 be the additive group in R p
. Define G = Hi © H 2 the direct sum
of H i and H 2 . The transformation g = (hi,k 2 ) € G transforms H 2 . The
transformation g — (fti,fta) € G transforms
X{ - t fti^i + h 3 , t = l , . . . , n ,
( M , E ) - * (ftiM + ha .f ti Sf ti ) .
The corresponding transformation on the sufficient statistic is given by (X,S)
- * (fti-A + / i 2 , ftiSftJ). A max im al invariant in the space of (X, S) under G is
R? = S12S22S21
and a corresponding maximal invariant in the parametric space of (p, E ) is p 2 .
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Equivariant Estimation in Curved Models 61
3.3.1. Characterization of equivariant estimators of £
Let S p be the space of all p x p positive definite matrices and let G £ ( p ) be
the group of pxp lower triangular matri ces wi th positive diagonal elements. A n
equivariant estimator d{X, S) of £ with respect to the group of transformations
G is a measurable function d(X, S) on S p xR p to S p satisfying
d(hX + £, hSh') = hd(X, S)ti
for all S £ S p ,h £ Hi, and X,£ € R p' From this definition it is easy to
remark that if d is equivariant with respect to G then d(X,S) = d(0,S) for
all X £ R P ,S £ S p . Thus without any loss of generality we can assume that d
is a mapping from S p to S p . Furthermore, if u is a function mapping S p into
another space Y (say) then d* is an equivariant estimator of u ( £ ) if and only
if d* — u • d for some equivariant estimator d of £ .
Let
and let G be the group of induced transformations corresponding to G on 0 P .
L e m m a 3.1. G acts transitively on 0 P .
Pr oof . It is sufficient to show that there exists a g — (A ,£ ) £ G wi th
ft £ Hi, ^ £ RP such that
1 1 P - 2 \ j N
(/i/t + £> A£A') =
1 p 0
p 1 0
.0 0 / /
1p - 2
(3.23)
/
If p = 0, i.e., £ 1 2 = 0, we take ha = £ , , 1 / 2 , f = - ft p to obtain (3.23 ). I f
p # 0, choose A n = E u
1 / 2
, A 2 2 = F £^3 , where T is a (p - 1) x (p - 1)
orthogonal matrix such that
£ n
1 / 2
E1 2
£2
-2
1 / a
r = 0 , 0, 0 ) ,
and £ = - A / / to obtain (3.23).
The following theorem gives a characterization of the equivariant estimator
d(S) of S.
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62 Group Invariance in Statistical Inference
T h e o r e m 3.5. An estimator d ofT. is equivariant if and only if it admits
the decomposition
a { S ,
~ {a^WR-iSu R~ 2 a 22 tR)S 21S,-1
1S1 2 + C(R){S22-S 2l S-1Sl2 )
(3.24)
where C{R) > 0 and
a /f,\ _ ( a-ii{R) a12(R) A [ K )
- \a 2l (R) a 22(R)
is a 2 x 2 positive definite matrix. Furthermore
d Y i di 2 d 2 2 <ki — p 2 — S ^ E i j i E j " ^ S a i
if and only if a n
l a i 2 a 2 2 a 2 \ = f> 2 •
N o t e . Th e d;j are submatrices of d as partitioned in (3.24) and aij =
aij(R).
P r oo f . T h e sufficiency part of the proof is computational. I t consists
in verifying d(hsh') — hd(S)h' for all h g H , S e Sp and d^,1 d^d^o1 d 2i —
<iii^i2a 22 a 2i. It can be obtained in a straightforward way from the compu
tations presented in the necessary part.
To prove the necessary part we observe that
P - ( R *)• FI >°<
and d satisfies
for all T € 0(p - 2). Th i s implies that
Ii 0
o r
P o \ _ / M R ) o0 I P- 2 ) { 0 C(R)I P_ 2
with C(R) > 0.
In general, S has a unique decomposit ion of the form
S =Sn Sl2 \ _(Ti 0 \ / l V \(T[ 0
Sn S 22 ) \ 0 T j \U V » J U T 2
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Equina Hani Estimation in Curved Models 63
with 7\ G G £ ( 1 ) , T 2 e G+(p - 1}, and U = T 2-1S 21T,-> . Without any loss of
generality we may assume that U ^ 0. Corresponding to U there exists a B G
0 ( p - 1) such that U'B = {R, 0, . . .0 ) with R = \\U\\= ( S ^1 S^S^1 5 2 1 ) = >
0. For p > 2, 5 is not uniquely determined but its first column is R~l U.
Using such a B we have the decomposition
1 W \ _ ( \ O W P 0 \ (l 0
U !„.,) ~ {0 Bj \ 0 J P _2 ) \0 B'
and
« - ( ? i ) - ( J S ) ( T £ ) ( ! * ° ) ( »:
i )
(n O f an(f i ) R - ' a u t R J l / ' V TJ 0
p - i - W ) A o 13
fi-'a^ffi)^! R- 2
a 2 2(R}S 2-l S-l 'Sl 2 + C(R){S 22 -S 2iS^Sl2 )
which proves the necessary part of the theorem. •
3.3.2. Characterization of equivariant estimators of 0
The following theorem gives a characterization of the equivariant estimator
of 0.
T h e o r e m 3.6 . If d* is an equivariant estimator of 0 then d*(S) has the
form
d*(S) = R'1 a(R)S 22
1S 2 , ,
where a : R+ —* R.
Proof. Define u:S p ^ RP'1 by
« ( £ ) = £ = £ - % ! .
If a" is equivariant then
d*(S) = (R-^WSnS^Su + C(R)(S 22 - S^S^S12 ))~l
S^R-'a^R)
= (T 2(R- 2
a 22(R)UU' + C f f l H C V ! - UU r
)T^S 2l R-v
a 21{R)
= R~l (a 22(R) + (1 - R 2
)(C(R))-1
a 21(R)S 2- 2
1
S 21
= R-1 a{R)S 2- 2
l S 2i • •
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64 Group Invariance in Statistical Inference
The risk function of an equivariant estimator d" of 0 is given by
R(0,d*) = Ex(L((3,d'))
= E L{S^(R-1 a(R)S nS 2-.i
l
- 0}'S 22{R~l a(R)S 22
l S 2l - 0)}
= E z{a?{R) - 2R-1
a{R)S-[ l l
S120 + S^0'S 220} . (3.25)
T h e o r e m . 3. 7. T h e best equivariant estimator of 0 given p, under the
loss function L is given by
R-l a*(R)S 2- 2
l S 2 , (3.26)
where
£ r ( n ± i + l ) r ( ^ - M ) ( V V ) 7 I T +
a'(R) = r p 2 ^ _ . (3.27)
£ r2 (2=1 + m ) ( p 2 r
2 ) m / m ! r (2=1 + m)
P r o o f . From (3.25), the minimum of R(0,d*) is attained when a(R) =
a*(R) — E^S^1
Si20R~1
\R). Since the problem is invariant and d" is equiv
ariant we may assume, without any loss of generality, that
- M f ,1)
with C(p) = \ \ £ j . To evaluate o*( fi) we write
5 2 2 = T T ' , T e G + ( p - l ) , r = (%)
5 2 1 = RTW, 0 < B < l , B
r
6 fl""
1
,S u = W W .
The joint probability density function of (R, W,T) is (Gir i (1977))
1 P - i i
i = 2 j = l
1 P _ l
* e x p {
~ 2 ( i - P
2
)( W i
"+
* » -2 T
P ^ W
^ n fe) •
i=i (3.28)
A straightforward computation gives (3.26). •
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Equivariant Estimation in Curved Models 65
L e m m a 3.2. Let 0 > 0, 0 < 7 < l , m be an integer.
Then
^Vja + m + l) T(0 + l)^
1=0T{a +1)
Proof .
^ r ( a + m + l ) r ( ^ + l ) j
i=0
d"
(=1
- 0
= (1 - 7 ) ^ ) ^ ( 1 + „ ) • + « - » ( l - ^
T h e o r e m 3.8. / / m — —p) is an integer then
•
a ' M ^ - ^ - l p y
i=0
m,
1 - r V
1 - r V
Pro of. Follows trivial ly from Lemma (3.2).
Note. If p 2 is such that terms of order (rp) 2 and higher can be ne
glected, the best equivariant estimator of 0 is approximately equal to p 2(n — 1)
(p-irl S 22
1S 2i. The mle of 0 is S 22S 2S .
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66 Group Invariance in Statistical Inference
E x e r c i s e s
1. Let X a = (Xai,- -,X aj ,)',a = 1,...,N( > p) be independently and iden
tically distributed p-variate normal random vectors with common mean fi
and positive definite common covariance matrix £ and let
show that if & = A V E - 1 / * is known, then NX'(S + NXX')~l X is an
ancillary statistic.
2. Let X a = (X ai,...,X ap y,a = 1,... ,2V( > p) be independently and identi
cally distributed p-variate normal random vectors with mean 0 and positive
definite covariance matrix £ and let
E _ (S
(12) \
\ £ ( 2 1 )
with £ ( l l ) : l x l . Given the value of p 2 - £ ( 1 2 , £ ( ~ 2 ) £ ( 2 1 1 / £ i i find the
ancillary statistic.
3. (B as u (1955)). If T is a boundedly complete sufficient statistic for the family
of distributions {Pg,9 g ft}, then any ancillary statis tic is independent of
T.
4. Find the conditions under which the maximum likelihood estimator is equiv
ariant.
Re fe r e n c e s
1. S. Amari, Differential geometry of curved exponential families - curvatures and
information loss, Ann. Statist. 10, 357-385, (1982a).
2. S. Amari, Geometrical theory of asymptotic ancillary and conditional inference,
Biometrika, 69, 1-17 (1982b).
3. D. Basu, On statistics independent of sufficient statistic, Sankhya 20, 223-226
(1955).
4. D. R. Cox and D. V, Hinkley, Theoretical Statistics, Champman and Hill, London,
(1977).
5. B. Efron, The geometry of exponential families, Ann. Statist., 6, 262-376, (1978).
6. B. Efron and C, Morris, Stein's estimation rule and its competitors, An empirical
approach, J . Amer. Statist. Assoc. 68, 117-130, (1973).
7. B. Efron and C. Morris, Stein's paradox in statistics, Sci. An n. , 239, 119-127,
(1977).
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Equivariant Estimation in Curved Models 67
8. R. A. Fisher, Theory of statistical estimation, Proc. Cambridge Phil. Soc. 22,
700-725, (1925).
9. N. Giri, Multivariate Statistical Inference, Academic Press, N.Y., 1977.
10. D. V. Hinkley, Conditional inference about a normal mean with known coefficent
of variation, Biometrika, 64, 105-108, (1977).
11. T. Kariya, N. Giri, and F . Perron, Equivariant estimator of a mean vector /i of Npifi, E ) with u"Z~l u = 1 or £"
V
V = c or £ = a 2(u'fi)I, J. Multivariate
Anal. 27, 270-283 (1988).
12. J . Kent, J . Briden, and K. Mardia, Linear and planer structure in ordered mul
tivariate data as applied to progressive demagnetization of palaemagnetic rema-
nence, Geophys. J. Roy. Astronom. Soc. 75, 593-662, (1983).
13. F . Perron and N. Giri, On the best equivariant estimator of mean of a multivariate
normal population, 3. Multivariate Analysis, 32, 1-16 (1990).
14. F . Perron and N. Giri, Best equivariant estimation in Curved Covariance Models,
40, 46-55 (1992).
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Chapter 4
SOME BEST INVARIANT TESTSIN MULTINORMALS
4 .0 . I n t r o d u c t i o n
In Ch apte r 3 we have dealt with some applications of invariance in statisti
cal estimat ion. We discuss in this chapter various testing problems concerning
means of multivariate normal distributions. Testing problems concerning dis
criminant coefficents, as they are somewhat related to mean problems will
also be cousidered. This chapter will also include testing problems concerning
multiple correlation coefficient and a rela ted problem concerning mult iple cor
relation with partial informations. We will be concerned with invariant tests
only and we will take a different approach to derive tests for these problems.
Rather than deriving the likelihood ratio tests and studying their optimum
properties we will look for a group under which the testing problem remain
invariant and then find tests based on the maximal invariant under the group.A justifi cat ion of this approach is as follows: If a testing problem is invariant
under a group, the likelihood ratio test, under a mild regularity condition, de
pends on the observations only through the maximal invariant in the sample
space under the group {Lehmann, 1959 p. 252). We find then the optimum
invariant test using the above approach, the likelihood ratio test can be no
better, since it is an invariant test.
4 .1 . T e s t s o f M e a n Ve c t o r
Let X = (Xi,...,X p ) be normally distributed with mean £ — E(X) =
(£i> — l i p ) 1 and positive definite covariance matrix £ — E(X — u) (X — p).
Its pdf is given by
68
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Some Best Invariant Tests in Mattinormals 69
fx(x) = {2 7 r ) -" / 2 (det exp j - i t r E "1
^ - 0 & - tf } • (4-1)
In what follows we denote 3: a p-dimensional linear vector space, and X', the
dual space of X. The uniqueness of nonmal distribution follows from the fol
lowing two facts:
(a) The distribution of X is completely determined by the family of distribu
tions olB'X, 9 £ X'.
(b) T he mean and the variance of a normal variable completely determines its
distribution.
For relevant results of univariate and multivariate normal distribution we refer
to Giri (1996). We shall denote a p-variate normal with pdf (4.1) as JVp
( f , S ) .
On the basis of observations x" — (x ai ,..., x ap ) , a — I,. ..,N from
N p{l, E ) we are interested in testing the following stat ist ica l problems.
P r o b l e m 1, To test the null hypothesis HQ: £ = 0 against the alternatives
Hi: £ j£ 0 when g, S are unk now n. This is known as Hotelling's T 2 problem.
Problem 2 . To test the null hypothesis HQ: £i — • • — £ p , — against the
alternatives H i : £ # 0 when £, £ are unknow n and p > p i .
Prob lem 3 . To test the null hypothesis Ha : £i — * * > — £ P l against the
alternatives H i : & = • - * = ^ = 0 when p, £ are unknown and pi + p 2 < p.
Let X - jj^X", S = E^(X a - X)(X a - X)'. (X,S) is sufficient
(minimal) for (£ , £ ) and \/NX is distributed independently of S as N p(y/ntl, E )
and S has Wishart distribution Wj,(JV—1, E ) with JI = J V - 1 degrees of freedom
and parameter E . Th e pd/ of S is given by
f X { d e t E ) ' n / 2 ( d e t s ) : i : : F 1 e x p { - | t r E - 1 a }
W p(n, £ } = < if s i s positive definite, (4.2)
I 0 otherwise.
Problem 1 remains invariant under the general linear group G;(p) transfer-
ing (X, S) -* (gX, gSg'), g G Gi(p). From Ex am pl e 2.4.3 with di = p
R =NX'(S + nXX')-1
X
NX'S-l
X
l+NX'S-tX
is a maximal invariant in the space of (X, S) under G ( (p) and 0 < R < 1.
(4.3)
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70 Group Invariance in Statistical Inference
The likelihood ratio test of HQ depends on the statistic (Gir i (1996))
T 2 = N(N -^X'S^X (4-4)
which is known as Hotelling's T 2 statistic. Since R is a one-to-one function of
T 2 we will use the test based on R. From Th eo re m 2.8.1 the pdf of i i depends
only on 6 — i V ^ ' E '1
^ and is given by
M > r ( p / 2 ) r ( ( 7 v - P ) / 2 ) 1 '
It may be verified that 8i is a maximal invariant under the induced group
(5j(p) = Gi(p) in the parametric space of (£ , E ) . Since £ is positive definite
tfi = 0 if and only if p = 0 and 6 > 0 for p ^ 0. F or invariant tests with respect
to G;(p) Problem 1 reduces to testing HQ: 6 = 0 against the alternatives Hi-
6 > 0. From (4.5) the pdf of R under HQ is given by
Mn) =
r ( a
J—
r M ( i - T j t l * - * & - *r ( f ) r ( V ) ( 4 . 6 }
if 0 < r < 1, 1 '
0 otherwise,
which is a central beta with parameters (|, ^j^)- From (4.5) the distribution
of R has monotone likelihood ratio in i i (see Lehmann, 1959). Thus the test
which rejects HQ for large values of R is uniformly most powerful invariant
(UMPI) under the full linear group for testing HQ against H i .
Since R ^ constant, implies that T 2 ^ constant we get the following theo
rem.
T h e o r e m 4.1 .1 . For Problem 1, Hotelling's T 2 test which rejects HQ
whenever T 2 ^ C, the constant C depends on the level a of the test, is UMPI
for testing HQ against Hi.
The group Gt{p) induces on the space of {X, S) the transformations
(X,S)^(gX,gSg'), 9 € G , ( p ) .
It is easy to conclude that any statistical test <p(X,S), a function of {X,S}
whose power function depends on ( / i , £ ) only through 6, is almost invariant
under Gj(p) . Since
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Some Best Invariant Tests in Multinomials 71
Ell ̂ (X <S) = E g .1)l _ g .lEg .l .ct>(X,S)
= Elti>2<KgX,gSg'),
thus
E^(<f>(X,S)-<P(gX,gSg')) = 0
for all ( / / , £ ) . Using the fact that the joint probability density function of
( X , S) is complete, we conclude that
<p(X 1S) = 4'(gX,gSg•}
for all g 6 <3J(P)> except possibly for a set of measure zero. As there exists a
left invariant measure (Example 2.1.3) on G ( ( p ) , which is also right invariant,
any almost invariant function is inva rian t (L eh man n, 1959). Hence we obtain
the following Theorem for Problem 1.
T h e o r e m 4 . 1 . 2 . Among all tests of HQ: £ — 0 with power function de
pending on 6, Hotelling's T 2 test is UMP:
P r o b l e m 2 . L e t Ti be the group of translations such that ti,g Ti trans
lates the last p — pi components of each X", a — 1 , . . . , N and let Gi be the
subgroup of Gi{p) such that g 6 G\ has the following form
g=(m o
\9{21) 9(22)
where g^U j is the p x x p x upper left-hand corner submatrix of g. This problem
is invariant under the affine group ( G ^ T i ) such that, for g £ Gi,tj € I i ,
(g, ( i ) transform
X a^gX a + tu a = l,..,,N.
A maximal invariant in the space of (X,S) under ( 0 i , T i ) is Bi, where
R ^ N X ^ S ^ + N X ^ X ' ^ r ' X ^ (4.7)
as defined in Example 2.4.3 with di == Jh- A corresponding maximal invariant
in the par ametric space of (£ , E ) under ( G i . T i ) is #i = N ^ E ^ . ^ J J . For in
variant test this problem reduces to testing H 0 : §i = 0 against the alternatives
Hi: &i > 0. From Theorem 2.8.1 the pdf of R t is given by
X
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72 Group Invariance in Statistical Inference
From (4.8) it follows that the distribution of R\ possesses a monotone likelihood
ratio in i%. Hence we get the following Theorem.
T h e o r e m 4 .1 .3 . For problem 2 the test which rejects Ho for large values
of Ri is UMPI under (Gi,Ti) for testing Ho against H\.
N o t e : As in Problem 1 this U M P I test is also the likelihood ratio test
for Problem 2. Using the arguments of Theorem 4.1.2 we can prove from
Theorem 4.1.3 that among all tests of Ho with power function depending on
Si, the test which rejects Ho for large values of Ri is U M P for Probl em 2.
Pr ob le m 3 . Let T 2
be the translation group which translates the last
p — pi — P2 components of each X a , a = 1 , . . . , N and let G 2 be the subgroup
of Gi(p) such that g e G 2 has the form
(9iu) 0 0 \
9 = 9(21) 3(22) 0 1
\9(31) 3(32) ff(33] /
where is pi x p t and <7(22] is p2 x p 2 . This problem is invariant under the
affine group (G2,T 2 ) transforming
X a
^gX a + t 2l a = l,..,,N
where g £ G 2 and t 2 € T 2 . A maximal invariant in the space of (X, S) under
(G 2 ,T 2 ) is C f l M b ) ) where
R l + R 2 - NXf2](Si22] + N J ? [ 2 ] A ' [ 2 ] ) "1
X [ 2 |
as defined in Ex am pl e 2.4.3 with di = pi, d 2 = p 2 .
A corresponding maximal invariant in the parametric space of (£,E) is
(61,62), defined by,
^ = ^ ( 1 ) ^ - ^ ( 1 ) .
Under the hypothesis H 6 ,$1 = 0, 6 2 = 0 and under the alternat ives Hi, fij =
0, 6 2 > 0. From Theorem 2.3.1 the joint pdf of Ri, R 2 is given by
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Some Best Invariant Tests in Multinomials 73
m f , _ n | N )
l
'2 > r t l p i j r c ^ r t ^ p - p ! - ^ ) )
From Giri (1977) the likelihood ratio test of this problem rejects Ho whenever
where the constant c depends on the level a of the test and under Ho, Z is
distributed as central beta with parameters (|(iV — pi — p 2 ) , \p?)-
From (4.9) it follows that the likelihood ratio test is not U M P I . However for
fixed p, the likelihood ratio test is approximately optimum as the sample size
N is large (Wald, 1943). Thus if p is not large, it seems likely that the sample
size commonly occurring in practice will be fairly large enough for this result
to hold. However, if the dimension p is large, it might be that the sample size
N must be extremely large for this result to apply.We shall now show that the likelihood ratio test is not locally best invariant
( L B I ) as S 2 - * 0. Th e L B I test rejects H 0 whenever
Ri + N ~ P l R 2 > c (4.11)
where the constant c depends on the level a of the test. Hotelling's T2
test
which rejects Ho whenever R, + R 2 ^ c, the constant c depends on the level O:
of the test, does not coincide with the L B I test, and hence it is locally worse
for Problem 3.
Definit ion 4.1.1. ( L B I test) . For testing H a : 9 € £ I H„, an invariant test
4>* of level a is L B I if there exists an open neighborhood fi i of fl/jo such that
Eebt?) > Ee{4>\ 0 e & - f i j i o , (4.12)
where 0 is any other invariant test of level a.
T h e o r e m 4.1.4. For testing H 0 : h*i = 0, tf 2 = 0 against H\ : S i = 0,6 2 >
0, the test which rejects H Q whenever R\ + N ~?' R 2 ^ c, where the constant c
depends on the level a of the test, is LBI as 6 2 —* 0.
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74 Group Invariance in Statistical Inference
P r o o f . Since (R\,R2) is a maximal invariant under the affine group
( G 2 , r 2 ) , the ratio of densities of (R1 ,R 2 ) under Hi, to that under HQ, is
given by,
m , ^ ) = 1 + 4 ( ' _ 1 + f i + i v ^ f 2 > ) + o ( , 2 )
f(fi,f 2 \0) 2 V P2
as 6 2 —* 0. Hence the power of any invariant test d> is given by
ES2($)=a-rE s ̂D $6 2 (-l + R i + ^ ^ R ^ j + o(6 2 )
as 62 —* 0, which is maximized by taking $ = 1 whenever Ri + N ~ Pl R 2
^ c. n
A s y m t o t i c a l l y B e s t Invar iant ( A B I ) T es t
Let # be an invariant test for testing H 0 : $1 = 0, 6 2 = 0 against Hi : S\ = 0,
6 2 = A with associated critical region fi = {(fi.fa) ! U{f\,ft) ^ c „ } satisfying
P « 0 ( f i ) = a;
P Hl (R) = 1 - e x p { - H ( A ) ( l + o < l ) ) } ;
f ! - " - 2 l f f ? = e x p W A J l G f A i + fifA)^,^)]}
+ fi(f l7f 2;A); (4.12)
where J B ( f i , f 2 ;A) — o(H(A)) as A —> 00 and 0 < ci < R(X) < c 2 < 00. Then
$ is an A B I test for testing HQ against H 1 as A —1 00.
T h e o r e m 4.1.5 . For testing HQ : S\ = 62 = 0 against H i : t>\ — 0 , £ 2 —
A > 0, Hotelling's test which rejects H 0 whenever Ri + R 2 ^ c, the constant c
depending on the level a of the test, is ABI as A —* 00 .
P r o o f . Since
_, , . , a a(a +1) x 2
= exp{ar(l + o( l) } as x —< 0 0 ,
we get from (4.9)
T ^ j ^ j = e x ^ " f i " M ( l + B(f u f2-,\))}; (4.13)
where S ( f i , f 2 ; A ) - o ( l ) as A -» 00 . It follows from (4.2) that the pdf of
U = R\ + R 2 satisfies
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Some Best Invariant Testa in Multinomials 75
P(U(Rl ,R 2 ) = R 1 + R 2 < c a )
= exp { ^ ( c « - 1 ) 0 + o ( l ) ) } • (4-14)
Thus, from (4.12) with H(X) = ~{c„ - 1), we conclude that Hotelling's T 2 test
is A B I . •
4 .2 . T h e C las s i f i c a t ion Pr ob le m ( T w o Popul a t ions )
Given two different p-dimensional populations Pi,P 2 characterized by the
probability density functions pi,p 2 respectively; and an observation x — [x\,
• . ., x PY the classification problem deals with classifying it to one of the two
populations. We assume for certainty that it belongs to one of the two pop
ulations. Denote by Oj, t = 1,2 that x comes from Pi and let us denote, for
simplicity, the states of nature Pi simply by i. Let L ( j , be the loss incurred
by taking the action a; when j is the true state of nature having property that
L(i ,a<) - 0, L(i,a,j) = ij > 0, j j£ i = 1,2,
Let (x, 1—T) be the a priori probability distribution on the states of nature.
The posterior probability distribution (£,1 — £), given the observation x, is
given by
_ rrp^x) ( ,5 «Pi(*J + { l
1
• '
The posterior expected loss of taking action al is given by
(1 - 7T)Ci P 2(l)
Kpi(x) + (1 - JT)P2(Z)
and for taking action a 2 is given by
irpi{x) + (1 - iv)p 2{x) '
From (4.16)-(4.17) it follows that we take action a 2 if
(4.16)
(4.17)
TC 2 Pl(x) , ( 1 - ^ ) C 1 P 2 ( X )
x p i ( x ) + (1 - 7 r ) p 2 ( i ) npi(x) + (1 - w)p 2(x)
and action aj if
7rc 2 pi(a;) > {1 - ir)clP2(x) ^
npi(x) + (1 - n)p 2(x) xp^x) + (1 - ?r)p2(i)
and randomize when equality holds.
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76 Group Invariance in Statistical Inference
(4.18) and (4.19) can be simplified as,
take action a 2 if > . , . — = K ( say) ,
take action a , if < JiT (4.20)
and randomize when — K.
Le t us now specialize the case of two p-variate normal populations wi th
different means but the same covariance matrix. Assume
Pi : W „ ( « , E ) ,
p 2 : A . p ( M , S ) ,
where a = . . . , ^ p ) ' , £ - (f t, . , - , £ p ) ' e i i p and £ is positive definite.
Hence
pa(«)= exp [f> - O'S"
1
^ HYX-H* - f*)] }
- exp ( s ' E ^ O - 0 4- | ( { ' E _ 1 £ - pE 'V )
Using (4.20) we take action a 2 if
and take action a x , otherwise. Th e linear functional T — £ - 1 ( p . — 4) is called
Fisher's discriminant function. In practice E , £, ft are usually unknown, and
we consider estimation and testing problems for T — ( r i , . . . , T p ) .
Let X a , a — 1 , . . . , Ni be a random sample of size A r i from N p(£, E) and
let Y a , a — 1 , . . . , A72 be a sample of size A72 from A r
p (p , E ) . Invariance and
sifficiency considerations lead us to consider the statistic
s = Y,{ xa - - + J2( Y ° - Y )( ya - y ) ' . a=l a=l
where N,X = X a , A ^ y = J T ^ Y° for the statistical inferences of V.
We consider two testing problems about T.
Problem 4 . To test the hypothesis H 0 : r p i + i = • • - — r p = 0 against the
alternatives Hi T ^ O .
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Some Best Invariant Testa in Multinomials 77
P r o b l e m 5. To test the hypothesis H 0 : T P 1 + 1 = • • • = T p — 0 against the
alternatives Hi : r p i + P 2 + i = • • • = T p = 0.
Without any loss of generality we restate our setup in the following canon
ical form, where X = {Xi,... ,X P ) is distributed as JV p (7 j ,E) and S is dis
tributed, independently of X, as Wishart W p
(n, E ) and T—
E
_ 1
J J . Since E ispositive definite, T = 0 if and only if n - 0. Thus the problem of testing the
hypothesis T = 0 against T ^ 0 is equivalent to Problem 1 considered earlier
in this chapter.
P r o b l e m 4, It remains invariant under the group G of p x p nonsingular
matrices g of the form
9=(m 0 ) (4 .21)
\9{21) 9(22) }where 9{\\) is of order pi, operating as
(X }S;T }S) ^ (gX^gSg'^g')-1?^^')
where X, jf are the mean and the sample covariance matrix based on N ob
servations on X. By Example 2.4 .3 a maximal invariant in the space of the
sufficient statistic (X, S) under G is given by (Ri^R^), with di = pi, d% = P2
and p — pi +P2 - The joint pdf of (r? i , R 2
) is given by (4 .4 ) where the expressions for 6i, 6
2 in terms of T, E are given by
Si + s 2 = A T ' s r ,
S 2 = Nr'(2)(x 22 rl r{2)
with £2 2
= ( E ( 2 2 ) - E ( 2 i ) E j_
n ) E ( 1 2 ) )_ 1
. Under the hypothesis H 0 61 > 0,
02 — 0 and under the alternatives HiSi > 0, i = 1,2. From (4 .4) it follows that
the probability density function of Ri under Ho is given by
» M r (4-22)
m\$i) =r ( i p , ) r ( i ( A / - p , ) )
x ( f i ) ^ p , " 1 ( l - f i ) ^ W " p l ) " 1
x exp | - ^ ! } 4> QjT, | M J I f i ^ l , (4.23)
and Ri is sufficient for t\.
Giri (1964) has shown that the likelihood ratio test for this problem rejects
Ho whenever
* = i T T 7 r ^ ( 4
-2 4 )
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78 Group Invariance in Statistical Inference
where the constant c depends on the level a of the test and under H 0 Z has a
central beta distribution with parameter {^{N - p ) , \p2) and is independent
oiRi.
To prove that the likelihood ratio test is uniformly most powerful similar
we first check that the family of pdf
UXH%)-h ^ 0 } (4.25)
is boundedly complete.
De fi ni ti on 4.2.1. ( B o u n d e d ly co mp le te ). A family of distributions
{Ps(x) '• 6 E fi} of a random variable X or the corresponding family of pdfs
{ps(x) : 6 e fi} is boundedly complete if
E6 h(X) = J k(x)dP s(x)
= j h{x)p s{x)dx
= 0
for all 6 6 fi and for any real valued bounded function h(X), implies that
h(X) = 0 almost everywhere with respect to each of the measures Pf.We also say AT is a complete statistic or X is complete.
L e m m a 4.2.1 . The family of pdfs { / ( f i ^ i ) : 6\ > 0} is boundedly com
plete.
Proof . For any real-valued bounded function h(R]} of Ri we get (using
(4.23))
E-Sl (h{Ri)) = j Hnmr^dfi
= e x P { - i * - 1 }
j - Jox
= exp <f l - 1 ^ I ^ 1 ) rl
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Some Best Invariant Tests in MultxnoTmals 79
where
h*(r1 )=h{r\)fi™-1(l-rl )u N -Ky-\
a g ( £ { P - P i ) , 5 ( * - P i ) + j )3
B ( i ( J V - p, ) , i P l + j ) B ( i ( J V - p), i ( p - P l ) )
and B(p , g) is the be ta function. Hen ce E g h(Ri) = 0 implies that
E - I / ^ C n ) ( n ) J ^ i = 0 - (4.26)
Since the right-hand side of (4.26) is a polynomial in 6\, (4.26) implies thatall its coefncents must be zero. In other words
/ h*(f\)f{dfi = 0 , j = 0 , 1 , 2 , . . . . (4.27) Jo
Let
where h"+
and ft*- denote the positive and negative parts of h'. Hence we get
from (4.27)
Jo Jo
for j= 0 ,1 , .. . ; which implies that
h*+(f 1 ) = h-(f l )
for all ^ except possibly on a set of measure 0. Hence we conclude that
h*(Ri) = 0 almost everywhere {Pg1 (Ri) : Si ^ 0} which, in turn, implies that
h(f i i ) = 0 almost everywhere. •
Since Ri is comple te, it follows; from Lehman n (1959) , p. 34; that any
invariant test 4>(Ri,R 2 ) of level a has Ne ym an Struct ure with respect to Ri,
i.e.,
ElIll (<p(Rll R 2 )\Ri ) = cl .
To find the uniformly most powerful test among all similar invariant tests
we need the ratio R
fH l (fl,f 2 \Rl = f i ) R =
}H a{T\,T 2 \Ri = f%)
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80 Group Invariance in Statistical Inference
where /^( f i f f^- f i i = r i ) denotes the conditional pdf of (RuRg) given Ri =
f i under Hi,i = 0,1. From (4.9) we get
H - exp | - i | 2 ( l - fx)} <t> ~ P i ) , \(P ~ P i ) ; - (4-28)
From (4.28), using the fact that Z is independent of R± when H 0 is true, we
get the following theorem:
T h e o r e m 4.2.1. For Problem 4, the likelihood ratio test given in (4.24) is
uniformly most powerful invariant similar.
P r o b l e m 5. It remains invariant under the group G of p x p nonsingular
matrices g of the form
/ V >0 0 \
9 = 3(21) 3(22) 0
\3(31) 3(32) 3(33)/
with 3(H) : pi x p2, 9(22) : P2 x p2 submatrices of g. A maximal invariant in
the space of (X,S) (as in Problem 4) under G is (Ri, R 2 , R3) as defined in
Example 2.4.3 with d\ = p i , d 2 = p 2 and d$ = p - pi - p2. B y Theorem 2.8.1
the joint probability density function of Ri, R 2 , Ri is given by
x f i " " - 1 ^ i » - l
F i ( r ^ - W - » J - l
x f l - n - ^ - f a j i ^ - p ' "1
l, '=1 J = l >>J J
where
and
"» - EpJ' CTQ = ° ' P3 = p - p i - p a •
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Some Beat Invariant Tests in Multinomials 81
&1 = i V f E j i i j l ^ i ) + S ( 1 2 | r i 2 ) + £(18)1(3) ) 'Sj~, j
* ( s ( i o r ( i ) + S ( i 2 ) r f ! ) + £ ( i 3 ] r ( 3 ) ) ,
A + 6 2 = f s ( n ) r ( i > + s c i 2 ) r ( 2 > + s ( i 3 ) r ( 3 ) V s - i
\ S ( 2 1 ) r ( l ) + £(22)^(2) + £(231^(3) / ' 2 2 '
x j /
E ( i i ) r ( 1 ) + £ ( 1 3 | r ( 3 ) - i - £ j l 3 ) r ( 3 ) \ ^
V E ( 2 i ) r ( 1 ) + £ ( 2 2 ] r ( 2 j + £ ( 2 3 ) r ( 3 ) J
S , = W r ; , 1 ( E ( 3 3 1 - ( ^ ; ) ' E r i . ( ^ ) , r ( 3 ,
Under H 0 , 61 > 0, 6 2 = 0, 6 3 = 0, and under H i , Si > 0, S 2 > 0, 6 3 = 0. From
(4.29) it follows that Ri is sufficient for S\ under H 0 . From Giri (1964) the
likelihood ratio test of this problem rejects H0
whenever
Z = 1
~ R i
;S a $ c (4.30)
1 - tii
where the constant c depends on the level a of the test and under H 0 Z is
distributed independently of H i as central beta with parameter (^(N — p x —
J>2). \V2)-
Le t <p(Ri,R 2 ,R 3 ) be any invariant level a test of H 0 against H i . From
Lemma 4.2.1 H i is boundedly complete. Thus <j> has Neyman Structure with
respect to R x . Since the conditional distribution of (Hi, R 2 , R3) given Ri does
not depend on c\, the condition that the level a test <j> has Neyman structure,
that is,
H H D ( 0 ( H 1 , H 2 , H 3 ) | H I ) - Q
reduces the problem to that of testing the simple hypothesis 6 2 = 0 against the
alternatives 6% > 0 on each surface Ri — f\. In this conditional situation the
most powerful level a invariant test of S 2 = 0 against the simple alternativeS 2 — 6 2>0\s given by,
r e j e c t H 0 whenever <h Q(JV - P l ) , i p z ; ^f 26%j ^ c, (4.31)
where the constant c depends on a. Since
tf.2 = ( l - H i ) ( l - Z )
and Z is independent of H i , we get the following theorem.
T h e o r e m 4 .2 .2 . For Problem 5 the likelihood ratio test given in (4.30) is
uniformly most powerful invariant similar.
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82 Group Invariance in Statistical Inference
4 . 3. T e s t o f M u l t i p l e C o r r e l a t i o n
Let X a , a = 1, ,N beN independent normal p-vectors with com mon mean
fi and common positive definite covariance matrix E and let NX = E%-,X a
,
S = £%=1(X a -X)(X a
-X)'. As usual we assume JV > pso that 5 is positive
definite with probability one. Partition E and S as
\ E ( 2 i ] E(22) / ' \ 5(21) S ( 2 2 ) /
where E ( 2 2 j , 5 ( 2 2 ] are of order p— 1. Let
2 _ s
( i 3 ) E ( 3 2 ) E ( 2 1 )
(u)"
= E
the positive square root of p 2 is called the population multiple correlation
coefficent.
P r o b l e m 6. To test H 0 ; p 2 = 0 against the alternative Hi : p 2 > 0. This
problem is invariant under the affine group (G,T), operating as
(X, 5; p, E ) - » ( g X + t, gSg'; gp +1, gXg')
where g g G i s a p x p nonsingular matrix of the form
g=f «")0 9(22)
where p( 2 2 ) is of order p - 1 and T is the group of translations t of components
of each X a . A maximal invariant in the space of {X,S) under the affine group
(G,T) is
r 2 _ ^ 2 2 > 5 ( 22)5(21)
5 ( H )
which is popularly called the square of sample multiple correla tion coefficent.
D i s t r ibut ion o f R 2 . To find the distribution of R 2 we first observe that
R 2 5 (12)S[! 2 )
S (21)
1 R
2
5 (U
) -
5
( i 2 ) £( Z 2 )
S( 2 1
)
From Giri (1977)
5(u) - 5 ( i 2 ) 5 , 2 L £ ( 2 1 )
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Some Best Invariant Tests in Multinomials 83
is distributed as X 2 N- P w ' t n
N - p degrees of freedom and is independent of
S(i2). 5( 22). Thus R 2 /(l - R 2 ) is distributed as
1 •S
( 1 2)5 { 22 |5(22)
* N - p 2(11) - E(12) S[ 22] S(21)
But the conditional distribution of
S ( 12) 5(22)2
\ / E ( i i } - S { 1 2 ) S ( 2 2 ) S ( 2 l )
given 5( 2 2 ) , is (p — l)-variate normal with mean
^{22)^(22)^2)
^ E ( u ) - E(i2)S ( 22)S(21)
and covariance / . Hence R 2 /(l - R 2 ) is distributed as
1 2 | / E ( 1 2 ) S ( 2 2 | 5 ( 2 2 ) E ( 2 2 ) S ( 2 1 )
X%- p
X p * v
E ( 1 1 ) - E d a j E ^ E f a i )
where Xj.(A) denotes a noncentral chisquare random variable with noncentrality
parameter A and k degrees of freedom. Also
s ( i 2 ) E ( 2 2 ) 5 ( 2 2 ) S | 2 2 | S ( 1 2 )
E(12)S(22)S(21>
is distributed as chisquare Xw—v Since
S(12)S(~22)S(21) p 2
E ( u ) - E ( I 2 ) E ( - 2 1 E ( 2 1 ) 1 -P 2 '
we conclude that R 2 /(l - R 2 ) is distributed as
,2
pl2 1 ^( 2 2 )^2 2 )^ ( 2 2 )^ (2 in
1 2 I _P_ 2 \
Using the fact that a noncentral chisquare random variable XmW can berepresented as xt1+2K> w n e r e K is a Poisson random variable with parameter
^, we conclude that
^ ( r r ^ - i )
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84 Group Invariance in Statistical Inference
is distributed as X - p _ 1 + 2 K - , where the conditional distribution of K given x ? v - i
is Poisson with parameter \[p 2l[l - P 2 )]XN-I - Let A/2 - i p 2 / ( l - p 2 ) . Then
P(K = k) =
fc!r<±(iv-i))
with lb = 0, 1 , . . . . Thus
25<w
-1
>r(i(Ar_i))A;!
_ r(§(iV-i)+*)A*
r ( | ( N - l ) + fc) , k _ 2 ) i ( N _ i , { 4
is distributed as X p
-1 + 2 K (4.34)
—T- l o U 1 M I 1 U U . c u d o t —
1 - B 2
X
2
N-
where K is a negative binomial with pdf given in (4.33). Simple calculations
yield
(N - p)R 2
(p~l)(l-R 2 )
has central F distribution with parameter (p - 1, N - p) when p 2 — 0. From
(4.33)-(4.34 ) the noncentral distribution of R 2 is given by
( j - r 2 ) l ( N - P - 3 ) ( r 2 ) l ( P - 3 ) ( 1 _ ftHtr-l)f R ' { r > - r ( i ( N - i ) ) r ( i ( i v - p ) )
, f . W W ( j ( f f - i ) + i) . . . . .x f e ^ ( i ( P - D + l)-
( 4 ' 3 5 )
It may be checked that a corresponding maximal invariant in the parametric
space is p 2 .
T h e o r e m 4.3.1. For problem 6 the test which rejects H 0 whenever R 2 ^ C,
the constant c depends on the level a of the test, is uniformly most powerful
invariant.
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Some Beat Invariant Tests in Multinomials 85
P r o o f . From (4.36)
~ ( A 2 ) T ^ ( J v - i ^ ) r ( j ( p - i ) )
^ i ! r ( i ( p - i ) + i ) r2
( i ( 7 v - i ) ) '
which for a given value of p 2 is an increasing function of r 2 . Using Neyman-
Pearson Lemma we get the theorem. •
As in Theorem 4.1.2 we can show that among all tests of Ho : p 2 = 0
against Hi : p 2 > 0 with power function depending only on p 2 the fi2-test is
UMP.
4 .4 . T es t of Mu l t ip l e Co rr e l at io n wi th P a r t i a l I n f o r m a t i o n
Let X be a normally distributed p-dimensional random column vector with
mean p and positive definite covariance matrix S, and let X a , a — 1,... ,7V
[N > p) be a random sample of size N from this distribution. We partition X
as X = (Xi,X^,X'^Y where X\ is one-dimensional, X^ is pi-dimensional
and X ( 2 ) is p2-dimensional and 1 + p i + P 2 = p. Le t p\ and p denote the multi
ple correlatron coefncents of X\ with X[ 2 ) and with (X'^X'^y, respectively.
Denote by p\ = p
2
— p\. We consider here the following two testing problems:
P r o b l e m 7. To test H m: p 2 = 0 against the alternatives Hi\; p\ — 0,
p\ = X > 0.
P r o b l e m 8. To test H 2 o- p 2 = 0 against the alternatives H 2 \: p\ = 0,
pl = X>0.
Let NX = X^,X 0 , S = S ^ ^ A - " - X ) ( X ° - X ) ' , 6 ( f J denote the i-vector
consisting of the first i components of a vector b and C[;j denote the i x i upper
left submatrix of a matrix c. Partitions 5 and E as
(5 n 5 ( 1 2 ) S ( 1 3 ) \ / E n E( i 2 j E ( 1 3 )
5(21] 5 | 2 2 ] 5 [ 2 3 ) I , E —I S(2i ) E ( 2 2 ) E ( 2 3)
5(31| 5(32 ) 5(3 3)/ \^(31) E ( 3 2 ) E ( 3 3 )
where 5(2 2
) and E(2
2 ) are each of dimension pi x pi ; 5 (3 3
) and E (3 3
j are eachof dimension P2 x p 2 . Define
Pi = E ( i 2 ) E ^ 2
1
2 ) E ( 2 1 1 / E 1 1 ,
p 2 = ^ + p I = { E , 1 2 ) E , 1 3 ) ) f g 2 2 ; ^ ^ " ' ( E d ^ ^ l ' / S n ,
\ (32) ^(33) /
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86 Group Invariance in Statistical Inference
Rl - £ ( i 2 ) S ( 2 2 ) S < 2 I ) / \ S i i ,
4 + fl2 = c% 3 )^ }) ( f p jjgj )
_ 1
•
It is easy to verify that the translation group transforming
( X , S ; i i , E ) ^ ( X + &,S;p + 6 , E )
leaves the present problem invariant and, along with the full linear group G of
p x p nonsingular matrices g
(gu o o \
9 = 0 ff(23) 0 (4.36) \ 0 9(32) 9(33) /
where ffii : 1 X 1, 9 { 2 2 ) : pi x pu gl33): p 2 x p 2 , generates a group which
leaves the present problem invariant. The action of these transformations is to
reduce the problem to that where p, = 0 and S = £%=1 X a X a' is sufficient for
E , where N has been reduced by one from what it was originally. We now treat
later formulation considering X", a = 1 , . . . , TV ^ p > 2, to have a common
zero mean vector. We therefore consider the group G of transformations g
operating as
(S,Z)^(gSg\gU)
for the invariance of the problem. A maximal invariant in the sample space
under G is {Ri,R 2 ) as defined in (4.11). Since S is positive definite with
probability 1 (as N ^ p ^ 2 by assumption): > 0, R 2 > 0 and Ri + R 2 =
R 2 , the squared sample multiple correlation coefficent between the first and the
remaining p - 1 components of the random vector X. A corresponding maximal
invariant in the parametric space under G is (p\,p\). T h e joint probability
density function of (R^R 2 ) is given by (see Giri (1979))
/ a ( n , f 2 ) - m - P 2
r N/2
(l ~ fi - f 2 )^ N
- p
~V T J f o ) * * - *
x
" A S S « w t * + A > — 1 9 , 1 • < 4
'3 7 )
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Some Best Invariant Tests in Multinormals 87
where
% = 1 - E P)w i t h = 1
'
ff
i = Ep
J'a
?=
~ f)/Wi-i.
.2
7 .
and i f is the normalizing constant.
By straightforward computations the likelihood ratio test of Hio when fl —
{ ( « , E ) : E13 —0} rejects J?io whenever
f ^ c , (4.38)
where the constant c depends on the size a of the test and under HM , RI has a
central beta distribution with parameter (i p i , ± ( N - p i ) ) , and the likelihood
ratio test of H 20 when fl = { (p , S) : S 1 2 = 0} rejects H 2 Q whenever
^ 1 ' ^ ~ ^ 4 C , (4-39)1 - r j
where the constant c depends on the size a of the test and under H 2 o the
corresponding random variable Z is distributed independently of R\ as central
beta with parameter (^(JV — pi — ps) , \p 2 )-
T h e o r e m 4.4 .1 . For problem 7 the likelihood ratio test given in (4.38) is
UMP invariant.
Proof . Under H i0 f t = 1 , i = 0,1,2. Hence a2
= 0, §i = 0, i = 1,2.
Under ff u P2
. = A, p\ = 0, % = 1, 71 = 1 - \ a 2 = 1 - A, 7 2 - 1 - A, al = 0,
81 = 4fiA and t?2 = 0. Thus
/ f l n ( n , r 2 ) _ ^ _ A j - w / 2
(801i=0
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88 Group /ntrarinnce in Statistical Inference
Using the Neyman-Pearson Lemma and (4.40) we get the theorem. •
T h e o r e m 4.4 .2 . For Problem 8 the likelihood ratio test is UMP invariant
among all tests <j>(Ru R.2) b^sed on R\,R\ satisfying
Proof. Under p\ = 0, p\'= A, % = 1 , % 1, 7a = 1 - A, &{ - 0,
8 a\ = A, h = 0 and 0 2 = 4 f 2 A(l - f i A ) - 1 .
Hence
fn 3 A f 2\T l)/fH 20(
f 2\n) = fH 1 A rl,?2)/fnio( rl,r2)
0 0 . r a ( J V - p 1 ) + t ) / / 4f 2 A
— (2i)! \ l - f 1 A / ( 4 4 1 )
From (4.41) / j y a j ( f 2 | r i ) has a monotone likelihood ratio in f 2 = (1 — z)(l —
fi). Now using Le hm ann (1959) we get the theorem. •
E x e r c i s e s
1. Prove Equations (4.3) and (4.5).
2. Prove (4.10).
3. Show that JrJ^(A) is distributed as X m+2K w n e r e i f is a Poisson random
variable with parameter jA.
4. Prove (4.31).
5. Let 7ri , ?r2
be two p-variate normal populations with means p\ and / i2
andthe same positive definite covariance matrix S . Let X — {Xi,... ,X P )' be
distributed according to ir\ or ir 2 and let b = ( & i , . . . ,b p )' be a real vector.
Show that
[E1lb'X)-E2(b , X)\ 2
var(b'X)
is maximum for all b if 6 — E- 1 (pi — p 2 ) where Ei denote the expectation
under Tij.
6. (Giri, 1994a). Let X a = {X al ,..., X op )', a = 1 , . . . , N(> p) be indepen
dently identically distributed p = 2p!-variate normal random vectors with
mean p = (/*!..- • tUp)' a n < * c o m m o n covariance matr ix E (positive definite).
Let X = jjX^X^ S = ^ , ( X a - X)(X a - X)'. Write
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Some Best Invariant Tests m Multinomials 89
E _ / S ( u ) £ ( i a ) \ s = ( 5 ( 1 1 ) % 2 } V
X = ( X i , . • • X p ) — ,
where S/yj and Sufy are pi x pi for all t, j and X( i) = ( X i , • • • , Xp
, ) '
(a) Show that the likelihood ratio test of H a : M(i) = p ( 2 j , with p =
{fi'w ,p[ 2) )\ p ( 1 } = ( M I I - - I M P I ) ' rejects rY 0 for large values of
T2 = N ( X ( 1 ) - X ( 2 ) ) ' ( 5 ( 1 1 ) + 5(22) - 5(i2) - 5(2i )) _ 1(-A"[i) - X ( 2 | )
where T 2 is distributed as x P 1 ( S 2 ) / ' x % - P 1 with 6 2 - /V (M ( I ) ~ r*(2])'
• (E ( i i ) + E (22) - E ( 2 1 1 - E ( 1 2 ) )_
' ( M ( I ) - "(2)) and X p , ( - ) . X w - P l
a r e
independent.
(b) Show that the above test is U M P I .
7. (Gir i , 19946). In problem 5 let T - E " V - ( T i r 2 p J ' = (T^j,Tfaf
with r ( 1 ) = ( r 1 [ . . . , r j „ ) ' .
(a) Show that the likelihood ratio test of HQ : T ( i ) = r ( 2 ) rejects Ho for
small values of Z
_ 1 + iV(X( i ) + X ( 2 ) ) , ( S ( n ) + 5(22) + 5(21) + 5 ( i 2 ) ) ~ 1 ( X ( i ) + X(2))
l + NX'S^X
where Z is distributed as central beta with parameter {^{N — pi), \pi)
under Hn.
(b) Show that it is U M P I similar.
(c) Find the likelihood ratio test of H 0 : T ^ , = AF( 2 ) when A is known.
8. (Gir i , 1994c). In problem 6 write
(£ ( i i ) E ( 1 2 j E ( 1 3 ) \ / S ( i t ) 5 ( 1 2 ) S ( 1 3 |
E(2i) E { 2 2 ) E(23) I , 5 = I 5(2 i) 5( 2 2) S ( 2 3 |
E(3l) E ( 3 3 ) E ( 3 2 ) / \ -S"<31) 5(32) 5(33)
where E ( 3 1 ) and 5(U
) are 1 x 1, E ( 2 2 ) and 5(2
2) are pi x pi and E ( 3 3
j and5(33) are pi x pi w it h 2pi = p — 1.
(a) Show that the likelihood ratio test of H 0 : E ( 1 2 ) = E ( 1 3 ) rejects H a for
large values of
R\ = ( 5 ( 1 2 ) - 5 ( i 3 ) ) ( 5 ( 2 2 ) + S ( 3 3 ) - S ( 3 2 ) - 5 ( 23 ) ) 1 (5(21) ~ 5 ( 3 ! ) ) / 5 ( U ) ,
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90 Group Invariance in Statistical Inference
the probability density function of R\ is given by
_ ( 1 - ^ - 0 ( 1 - ^ ( ^ - 3 )
r ( i ( i v - i ) ) r ( i ( i v - P l ) )
j = 0
( ^ ) J
( ^ ? >P l + J
" -1 )
r 3
( ^ ( J v - i ) + j) mhpi+j)
where
p 2 = i V ( £ ( 1 2 ) - 2(13)1(2(22) -I- £ { 3 3 ) - £ ( 3 2 ) - E ( 2 3 ] ) ~l
x ( £ ( 2 i ) - £ ( 3 i ) ) / £ ( 1 1 ) .
(b) Show that no optimum invariant test exists for this problem.
R e f e r e n c e s
1. N, Giri, Locally Minimax Tests of Multiple Correlations, Canad. J . Statist. 7,
53-60 (1979).
2. N. Giri, Multivariate Statistical Inference, Academic Press, N.Y., 1977.
3. N.Giri, On the Likelihood Ratio Test of a Normal Multivariate Testing Problem,Ann. Math. Statist. 35, 181-189 (1964).
4. N. Giri, On a Multivariate Testing Problem, Calcutta Stat. Assoc. Bull. 11, 55-60
(1962).
5. N.Giri, On the Likelihood Ratio Test of A Multivariate Testing Problem II, Ann.
Math. Statist. 36, 1061-1065 (1965).
6. N. Giri, Admissibility of Some Minimax Tests of Multivariate Mean, Rapport de
Research, Mathematics and Statistics, University of Montreal, 1994a.
7. N. Giri, On a Test of Discriminant Coefficients, Rapport de Research, Mathe
matics and Statistics, University of Montreal, 1994b.
8. N. Giri, On Tests of Multiple Correlations with Partial Informations, Rapport
de Research, Mathematics and Statistics, Univerity of Montreal, 1994c.
9. N. Giri, Multivariate Statistical Analysis, Marcel Dekker, N. Y . , 1996.
10. J . K . Ghosh, Invariance in Testing and Estimation, Tech. report no Math-Stat
2/672. Indian Statistical Institute, Calcutta, India, 1967.
11. E . L . Lehmann, Testing Statistical Hypotheses, Wiley, N.Y., 1959.
12. A. Wald, Tests of Statistical Hypotheses Concerning Parameters when the Num
ber of Observations is Large, Trans. Amer. Math. Soc, 54, 426-482 (1959).
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Chapter 5
SOME MINIMAX TESTS INMULTINORMALES
5 .0 . I n t r o d u c t i o n
The invariance principle, restric ting its attention to invariant tests only,
allows us to consider a subclass of the class of all available tests. Nat ur al ly
a question arises, under what conditions, an optimum invariant test is alsooptimum among the class of all tests if such can at all be achieved. A power
ful support for this comes from the celebrated unpublished work of Hunt and
Stein, popularly known as the Hunt-Stein theorem, who towards the end of
Second World Wa r proved that under certain conditions on the transforma
tion group G , there exists an invariant test of level a which is also minimax,
i.e. minimizes the maximum error of second kind (1-power) among all tests.
Though many proofs of this theorem have now appeared in the literature, the
version of this theorem which appeared in Lehmann (1959) is probably closein spirit to that originally developed by Hu nt and Ste in. P i tt man (1939) gave
intuitive reasons for the use of best invariant procedure in hypothesis test ing
problems concerning location and scale parameters. Wald (1939) had the idea
that for certain nonsequential location parameter estimation problems under
certain restrictions on the group there exists an invariant estimator which is
minimax. Peisakoff (1950) in his P h . D . thesis pointed out that there seems to
be a locuna in Wa ld 's proof and he gave a general development of the theory
of minimax decision procedures invariant under transformation group. Kiefer
(1957) proved an analogue of the Hunt-Stein theorem for the continuous and
91
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92 Group Invariance in Statistical Inference
discrete sequential decision problems and extended this theorem to other de
cision problems. Wesler (1959) generalized for modified min im ax tests based
on slices of the parametric space.
It is well-known that for statistical inference problems we can, without any
loss of generality, characterize sta tis tic al tests as functions of sufficient statis ticinstead of sample observations. Such a characterization introduces considerable
simplifications to the sample space without loosing any information concerning
the problem at hand. Though such a characterization in terms of maximal
invariant is too strong a result to expect, the Hunt -Stein theorem has made
considerable contribution towards that direction. T h e Hunt-Ste in theorem
gives conditions on the transformation groups such that given any test <f> for
the problem of testing H : 8 € fl// against the alternatives K : $ € fl/f, with
fi / / n fi/f a null set, there exists an invariant test such that
sup ES(f>> sup Ee yj, (5.1)eefi„ eer2„
inf Ee d> < inf Eeil>. (5.2)fe n* ~ eefi K
In other words, V behaves at least as good as any <f> in the worst possible cases.
We shall present only the statements of this theorem. Fo r a detai led discussion
and a proof the reader is referred to Lehmann (1959) or Ghosh (1967).
L e t V — {Pg,0 e f!} be a dominated family of distr ibutions on the E u
clidean space (X,A), dominated by a cr-finite measure p. Let G be the group
of transformations, operating from the left on X, leave fi invariant.
T h e o r e m 5.0.1. (Hunt-Stein Theorem). Let B be a a-field of subsets of
G such that for any A E A, the set of pairs (x,g) with gx £ A is in A x B
and for any B € B, g G G , Bg e B. Suppose that there exists a sequence of
distribution functions v n on ( G , B) which is asymptotically right invariant in
the sense that for any g € G , B € B
\im^\v n(Bg) - v»(B)\ = 0. (5.3)
Then, given any test <j>, there exists a test V1 which is almost invariant and
satisfies conditions (5.1) and (5.2).
It is a remarkable feature of this theorem that its assumptions have nothing
to do with the statistical aspects of the problem and they involve only the group
G. However, for the problem of admissibility of statistical tests the situation is
more complicated. If G is a finite or a local ly compact group the best invariant
test is admissible . For other groups the nature of V plays a dominant role.
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Some Minimax Test in Muttinormates 93
T h e proof of Th eo re m 5.0.1 is straightforward if G is a finite group. Let m
denote the number of elements of G. We define
As observed in Chapter 2, invariant measures exist for many groups and they
are essentially unique. But frequently they are not finite and as a result they
cannot be taken as a probability measure.
We have shown in Chapter 2 that on the group 0{p) of orthogonal matr ices
of order p an invariant probability measure exists an d this group satisfies the
conditions of the Hunt -St ein theorem. T h e group GT(P) of nonsingular lower
triangular matrices of order p also satisfies the condit ions of this theorem (see
Lehmann, 1959, p. 345).
5.1. L o c a l l y M i n i m a x T e s t s
Let (X,A) be a measurable space. Fo r each point (5, JJ) in the parametric
space f!, where 6 > 0 and ij is of arbitrary dimension and its range may depend
on i5, suppose that p{-; 6 ,n) is a probability density function on (3Z ,A) with
respect to some u-finite measure p . We are interested in testing at level a
(0 < a < 1) the hypothesis HQ : 6 — 0 against the alternative Hi : 6 = A,
where A is a positive specified canst ant and in giving a sufficient condition for
a test to be locally minimax in the sense of (5.7) below. This is a local theory
in sense that p(x;A, JJ) is close to p(x;o, 7?) when A is small. Obv iously, the n,
every test of level a would be locally minimax in the sense of trivial criterion
obtained by not substracting a in the numerator and the denominator of (5.7).
It may be remarked that our method of proof of (5.7) consists merely
of considering local power behaviour with sufficient accuracy to obtain anapproximate version of the classical result that a Bayes procedure with constant
risk is mi nima x. A result of this type can be proved under various possible
types of conditions of which we choose a form which is more convenient in m any
applications and stating other possible generalizations and simplifications as
remarks. Throughout this section expressions like o(A), o(/i(A)) are to be
interpreted as A — > 0.
For fixed a , 0 < a < 1 we shall consider critical regions of the form
(5.4)
where V is bounded and positive and has a continuous distribution function
for each {S,Tj), equicontinuous in (6 ,7)} for 6 < some 6$ and which satisfies
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94 Group Invariance in Statistical Inference
(5.5) P x ,v(R) = a + k(X)+q(X,V )
where q(X,n) = o{h(X)) uniformly in v with h(X) > 0 for A > 0 and h(X) =
o( l ) . We shall be concerned with a priori probability density functions £<J,Aand t]\,x on sets 6 = 0, 8 = A respectively, for which
{ f r o i f ^ = 1 + WW) + r(Xmm + B( X , A) (5.6)
where 0 < Ci < r(A) < c 2 < oo for A sufficiently sm all an d p(A ) = o ( l ) ,
B(x,X) = o(h(X)) uniformly in x.
T h e o r e m 5.1.1. (Locally Minimax Tests) If the region R satisfies (5.5)
and for sufficiently small X there exist £o,i. and fa^ satisfying (5.6) then R is
locally minimax of level a for testing Ho '• 8 = 0 against Hi : 8 = X as A —* 0,
that is to say,
lim inf , PUR)-o = t _ « -
i -o Sup0 A e Q a inf , Px, n{4>\ rejects i f 0 } - o
tu/iere Q Q is t/ie class of all level a tests <p\.
Proof . Let
r\ = [2 + h(X){g(X) + c a r ( A ) } ] - 1 .
Then
—p- = 1 + h{X) \g(X) + c a r(X)}. (5.8)
Using (5.7) and (5.8) the Bayes critical region relative to the a priori distribu
tion
£\ = (1 - T A ) £ O , X + n £ M
and ( 0 — 1 ) loss is given by
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Some Minimal Teat in Multino-rmales 95
Define for any subset A
Pi,*(A) = J Px,vW) 6 , A W ,
Vx=R — B\,
W X = B X - R . (5.10)
Using the fact that
sup B(x,X)
h{\)- o ( A )
and the continuity assumption on the distribution function of U we get
Plx(Vx + W x )=o(X). (5.11)
Also for U x = V k or
Pi\x(Ux) = PS,x(Ux)[l + Pim)}] - (5-12)
Let
r* x(A) = {1 - n) ^,A
£4 + n ( l - pfc(jf)).
Using (5.9), (5.10) and (5.11) the integrated Bayes risk relative to $\ is given
by
r'x(Bx) = rl(R) + (1 - n ) [ P 0 ' , x ( ^ ) - P 0 * A ( V A ) ]
+ n [ f , r , A ( ^ ) - F 1 * , A ( - y - A ) ]
= r A ( R ) 4- (1 - 2 r A ) [ P 0 * , ( ^ ) - P Q * A (V A ) ]
+ ^ O % ( V X + W A ) O ( M A ) )
= r A ( R ) + o( / l (A)). (5.13)
If (5.7) were false one could by (5.5) find a family of tests {0>J of level a such
that d> x has power function cv + g(X, ij) on the set 6 — A with
lim sup [inf g(A, JJ) - h(\)]/h(\) > 0 .
The integrated risk r' x of ^> with respect to £\ would then satisfy
contradicting (5.13).
l im s u P K ( i i ) - r , ] / h ( A ) > 0 ,A—0
•
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96 Group invariance in Statistical Inference
R e m a r k s .
(1) Let the set {6 = 0} be a single point and the set {8 = A } be a convex
finite dimensional Euclidean set where in each component $ of TJ is
00(A))- If
P1
? ' ^ = l + h(X) U(x) + T, 6i(*) OiifA) m + B{x, A, TJ) (5.14)p(ar;0,Jj)
where sif ay are bounded and sup X i , J3(X,A,JJ) = o ( / i (A) ) , and if there
exists any satisfying (5.6 ), then the degenerate measure f ? A which
assigns all its measure to the mean of £I,A also satisfies (5.6).
(2) T h e assumption on B can be weakened to Px,„{\B(x, A ) | < e k(X)} -» 0as A - * 0 uniformly in n for each e > 0. If the i are independent of
A the uniformity of the last condition is unnecessary. T he boundedness
of U and the equicontinuity of the distribution of U can be similarly
weakened.
(3) T he conclusion of Theorem 5.1.1 also holds if Q„ is modified to include
every family {<j>x} of tests of level a + o(h{X)). One can similarly
consider the optimality of the family {Ux} rather than single U by
replacing R by Rx - {x : U x{x) > c a ,x}, where PQt „{Rx} = a - qx(v)
with qxiv) — o(h{X)).
(4) We can refine (5.7) by including one or more error terms. Specifically
one may be interested to know if a level a critical region R which
satisfies (5.7) with
inf Px, n{R) = a + ci A + o(A), as A - > 0
also satisfies
l i m mUPxMRl-a-dX = ^
o sup^ e Q O inf , Px,v{<j> rejects H 0 } - a - c,X x
In the setting of (5.14) this involves two moments of the a priori distribution
£ I ,A rather than just one in Theorem 5.1.1. As further refinements are invoked
more moments are brought in.
The theory of locally minimax test as developed above and the theory of
asymptotically minimax test (far in distance from the null hypothesis) to be
developed later in this chapter serve two purposes. F i r s t the obvious point of
demonstrating such properties for their own sake. But well known valid doubts
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Some Minimax Test in Multinormales 97
have been raised as to meaningfulness of such properties. Secondly, then, and
in our opinion more important, these properties can give an indication of what
to look for in the way of genuine minimax or admissibility property of certain
tests, even though the later do not follow from the local or the as ym ptot ic
properties.
E x a m p l e 5.1.1. (T 2 -t es t) . Conside r Prob lem 1 of Cha pte r 4. L et X i , . . . ,
XN be independently and identically distributed J V p ( p , E ) random vectors.
Write NX = X „ 5 - £ f ( X ; - - X)'. Let 6 = > 0.
For testing H 0 : 8 = 0 against the alternatives Hi : 8 > 0 the Hotelling's T 2
test which rejects H Q whenever R - NX'{S + NXX')~l
X > c, where c is
chosen to yield level a, is U M P I (Theorem 4.1.1).
Note. For notational convenience we are writing R\ as R.
L et 6 — A > 0 (specified). We are concerned here to find the local ly
minimax test of Hn against Hj : S = A as A -> 0 in the sense of (5.7 ). We
assume that N > p, since it is easily shown that the denominator of (5.7) is
zero in the degenerate case N < p. In our search for locally minimax test as
A—> 0 we may restrict attention to the space of sufficient s tat isti c ( X , S). T he
general linear group G)(p) of nousingular matrix of order p operating as
( £ , s ; p , E ) — . (gx,gsg';gp, gT,g')
leaves this problem invariant. However, as discussed earlier (see also James
and Ste in, 1960, p. 376), the Hu nt -Ste in theorem cannot be applied to the
group G ( (p ) , p > 2. However this theorem does apply to the subgroup Gj- (p)
of nonsingular lower triangular matrices of order p. Thus, for each A, there
is a level a test which is almost invariant and hence for this problem which
is invariant under Grip) (see Lehmann, 1959, p. 225) and which minimizes,
among all level a tests, the minimu m power under H i . From the local point
of view, the denominator of (5.7) remains unchanged by the restriction to GT
invariant tests and for any level a test d> there is a G j invariant level a test
4>' for which the expression in f, Px , , [4>' rejects H a ) is at least as large, so
that a procedure which is locally minimax among GT invariant level a tests,
is locally minimax among all level a tests.
In the place of one-dimensional maximal invariant R under G j( p) we obta in
a p-dimensional maximal invariant (Ri,...,R p ) as defined in Theorem 2.8.1
with k = p and di — 1 for all i, satisfying
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98 Group Invariance in Statistical Inference
J2 Rj = NX{ A{S M + N X ^ r ' X ^ , i = 1, .. - ,p (5.16)
i=]
with R, > 0, £ ] ? = l flj - J? < 1 and the corresponding maxi mal invariant on
the parametric space of ( /* ,£) is (^ 6 P ) (Theorem 2.8.1) where
(5.17)
with Sf_i — 5. The nuisance parameter in this reduced setup is n =
(rn,...,%)', with jji = 6,/6 > 0, V _ ^ = 1 TJJ = 1 and under H 0 i) = 0. From
Theorem 2.8.1 the Lebesgue density of Ri,... ,R P on the set
i n , . . . , r p ) : r , > 0, J P f l < 1
is given by
/ ( r i , . . . , r „ ) =
x exp
y j , / J V - « + 1 i nSi
2' 2(5.18)
We now verify the validity of Theorem 2.8.1 for U — Yl^=i ^ - s i n c e those
preceding (5.5) for the locally minimax tests are obvious. In (5.5) we take
fi(A) = b\ with 6 a positive constant. Of course P\,„(R) does not depend on
77. From (5.18) we get with r = ( r i , . . . r p ) ' ,
/ o , , (r ) 2+ B(J,X,V) (5-19)
where B(r,\,-n) — o(A) uniformly in r , n as A —* 0. The equation (5.6) issatisfied by letting £ 0 | A give measure one to the single point TJ — 0 while
gives measure one to the single point rj = TP* = (TJ", . . . where
T?; - (N - j)~ L(N - j + l r v ^ o v - p ) , j = 1 , . . . , p
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Some Minimax Tesi in MultinoTmales 99
so that Y,i>3 n* + (N - j + l }rj; = £ for al l j. From Theorem 5.1.1 we get
the following theorem.
T h e o r e m 5.1.2. For every p,N,a{N > p) Hotelling's T 2 test based on
T 2 or equivalently on R is locally minimax for testing HQ : 6 — 0 against the
alternatives Hi : 6 = X as X —* 0.
E x a m p l e 5.1.3. Consider Problem 2 of Chapter 4. From Example 5.1.2
it follows that the maximal invariant in the sample space is ( J ? i , . . . , R P1 ) with
Ri = a n d t n e corresponding maximal invariant in the parametri c space
is ( f i i , . . . , (J P , ) with 8\ = Yii' 6j. Under Ho, Si = 0 and under Hi 6i — X.
Now following E xample 5.1.2 we prove Theorem 5.1.3.
T h e o r e m 5.1.3. For every pi,N,a the UMPi test which rejects Ho for
large values of R\ is locally minimax for testing Ho : &i — 0 against the
alternative Hi : 6\ — X as X —* 0.
E x a m p l e 5.1.4. Consider Problem 3 of Chapter 4. We are concerned here
to find the locally minimax test of Ho : Si = 0, S 2 = 0 against the alternatives
Hi : 8i = 0, 6 2 — X > 0. As usual we assume that N > p the determinator
of (5.7) is not zero. To find the locally minimax test we restrict attention
to the space of sufficient statistic (Xi S). The group (G2,T 2 ) operating as
(X, S) —* (gX + t 2 , gSg'), g € G 2 , t 2 € T 2 , which leaves the problem invariant,
does not satisfy the condition of the Hunt-Ste in theorem. However this theorem
holds for the subgroup (GT {p\ + p 2 ) , T 2 ). A maximal invariant in the sample
space under this subgroup is Ri...., R Pl + P I such that
^R^NX'u (SM + NXMX {i] )-l X [{l , i = l , . . . , p i + P 2 (5.20)
with Ri >0,Ri= Rj, Ri+R 2 = E ^ L i " 2 R 3 , and the corresponding
maximal invariant in the parametric space is 61,..., 6 Pl + P 2 such that
with Si > 0,6 = E j P L i 6j, h + 6 2 = 5,. Under H 0 Si = 0 for all i and
under Hi Si — 0, i — 1 , . . . , p i , fij = 6 = \ > 0. The nuisance parameter in this
(5.21)
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100 Croup Invariance in Statistical Inference
reduced set up is fj = ( r f i , . . . , Vp,+piY w ' * h if, = ^ . Under H 0 n - 0 and under
Hi, ifc = 0 i = l . . . - , p i , ifc > 0, « = pi + 1 P i + Pa with E f ^ + i * = L
Equation (5.19) in this case reduces to
/ o , „ ( r ) 2- l + f,-!- M ^ r j . + f / V - . j + l ) ^ + £ ( r , n , A )
(5.22)
as A —» 0 with B(r, JJ, A) = o(A) uniformly in r, n.
The set £ 2 = 0, 6i = 0 is a single point n = 0, so £O,A assigns measure
one to the single point ij = 0. Since the set {Si = 0, 6 2 — A (fixed)} Is a
convex p 2-dimensional Eucl idean set wherein each component jj ; is o(/ i(A)),
any probabi lity measure can be replaced by the degenerate measure ££ A
which assigns measure one to the mean (0,0,*7p,+i>-• • < n p,+ P2 °f &,*)•
Hence from (5.22)
/ fx, n(r) £ux(dv)
I fo.n(r)
= I + ^ - i + f i + "E *(l>+Cw-i+i>%J
+ B ( r , A , 7 j ) \iiAdn)
= 1 + - - l + f , + £ ^ ( ^ i t f + t W - j + l h n + B ( r , A )
L J = P 1 +
' V , > J / J (5.23)
where B{r, A) — o(/i (A)} uniformly in r .
Let il* be the rejection region defined by
Rk = {X : U(X) = J * , + kR 2 > C Q } (5.24)
where fc is chosen such that (5.23) is reduced to yield (5.6) and the constant
C a depends on the level a of the test for the chosen fc. Choose
( j y - j - i ) . . . ( / V - P l - p 3 ) ( A 7 - P i )
P2(iV - p , - P 2 + 1)
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Some Minimal Test in Muttinormales 101
j = pi + 1, . . . ,p i + p 2 - 1 ,
n _ ( ^ - P i ) ( 5 2 5 1
so that
N - p ,V M ^ - J + l l ^ ^—^-, J = Pi + 1, - - ,pl + P 2 •
The test 0* with rejection region
J T = j * : U ( X ) = % + > C „ | (5.26)
with Pa,\{R*) — a satisfies (5.6) as A —> 0. Furthermore any region R& of
structure (5.24) must have k — to satisfy (5.6) for some From
Theorem 4.1 .4, for any invariant region R*, Px tV (R*) depends only on A and
hence from (5.5) q{\,r)) — 0 and thus tp* is L B I for this problem as A —* 0.
Hotelling's T 2 test based on Ri + R 2 does not coincide with 4>' and hence it
is locally worse. It is easy to verify that the power function of Hotelling's test
which depends only on 03, 6\ being zero, has positive derivative everywhere,
in particular, at 8 2 — 0. Thus from (5.5), with R — R*, h(A) > 0. Hence we
get the following theorem.
Theorem 5.1 .4. For Problem 3 the L B I test 0* is locally minimax as A —* 0.
R e m a r k s . Hotelling's T 2 test and the likelihood ratio test are also invari
ant under (G 2 , T 2 ) and therefore thier power functions are functions of 6 2 only.
From the above theorem it follows that neither of these two tests maximises
the derivative of the power function at S 2 = 0. So Hotelling's T 2 test and the
likelihood ratio test are not locally minimax for this problem.
E x a m p l e 5.1.5. Consider Problem 6 of Chapter 4. The affine group
( G , T), transforming
( X , S; p, S ) (gX + t, gSg'; ga + t, gBg')
where g € G, t € T leaves the problem of testing H^p 2
— 0 again H\ : p 2
=
A > 0 (specified) invariant. However the subgroup ( G T ( P ) , T ) , where Gj-(p)
is the multiplicative group of nonsingular lower triangular matrices of order
p whose first column contains only zeros expect for the first element, satisfies
Hunt-Stein conditions. The action of the translation group T is to reduce the
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102 Group Invariance in Statistical Inference
mean a to zero and S — E a = i X a(X°)' is sufficient where N has been reduced
by one from what it was originally. We treat the latter formulation considering
A T 1 , . . . , A T W to have zero mean and positive definite covariance matrix E and
consider only the subgroup Gx{p) for invariance. A maximal invariant in the
sample space under Gx{p) is R= (R 2 , • - -, Rp)', where
_ S [l2 )[i] 5 ( ~ 2 ) [ i ] S ( 2 ] ) [ i ]
5 ( H )i = 2 , . . .,p (5.27)
i=i
where C[i] denotes the upper left-hand corner submatrix of C of order i and
tyi] denotes the i-vector consisting of the first i components of a vector 6 and
\
A
( 2 1 ) 3(22)7where 5(22) is a square matr ix of order (p— 1). As usual we assum e tha t N >p
which implies that 5 is positive definite with probability one. Hence Ri > 0,
R 2 = U = ^2*- 2 Rj S 1- (R 2 is the squared sample multiple correlation
coefficient). T h e corresponding ma xim al invarian t in the par ametr ic space is
A = (6 2 ,...,S PY, where
5 > -i-2
.2 _ V P
£('2)['] S (22)[i1 S<31)[i]
E ( U )
(5.28)
with 6, > 0, p 2 = E ? = 2 5 > - L e t
The joint distribution of J ? 2 , . . . , R p is (see Giri and Kiefer, 1964)
( l - X ) ^ ( l - T , ^ r i ) i l N - p - l )
nHr-Dr(l(N-p+ I ) ) ( l + E P
= 2 r j i C l - A ) / 7 i - 1])
x f i + i)^ N -+2] r(l{N -i +2))J= 2
ft,=D /3P=D \j=2
3=2 L
r ( i ( j v - i + 2) + ft; 4 T - J ( l - A ) / 7 j ( l + 7 r -1 0
(5.29)
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Some Minimax Test in Multinormates 103
where 7_,- = 1 - A J j ^ l f t , ttj = Sjfy. Th e expression 1 / ( 1 + TT̂ -
6j - 0. From (5.29) we obtain
means 0 if
/o,o(r) 2 3=2 \ i>J
+ B ( r , A , » / ) , (5.30)
where £J(j", A,T?) —o (A) uniformly in r, 77.
From Th eo re m 4.3.1 it follows th at the assumptio n of Th eore m 5.1.1 are
satisfied with U = £ P
= 2 -^j = R 2 a i m M-M = bX, b > 0. As in exam ple 5.1.3
letting £ i | A give measure one to n" = ( ) )§ , . . . . ,1(2} where j j j — (JV— j + l j - ' ( W —
j + 2 ) - 1 ( p - l ) _ 1 J V ( i V - p + 1), j = 2 , . . . , p - l we get (5.6) . Hence we prove
the following theorem.
T h e o r e m 5.1 .5 . For ever?/ p,N and a the R? test which rejects H 0 for
large values of the squared sample multiple correlation coefficient R 2 is locally
minimax for testing HQ against Hi : p 2 = A —> 0.
R e m a r k s . It is not a coincidence that (5.19) and (5.29) have the same
form. (5.18) involves the ratio of noncentral to central chisquare distribution
while (5.29) involves similar ratio with random noncentrality parameter. The
first order terms in the expressions which involve only mathematical expecta
tions of these quantities correspond each other.
E x a m p l e 5. 1. 5 . Consid er problems 7 and 8 of Sec. 4.3. T h e group G
as defined there does not satisfy the condition of the Hunt-Stein theorem.
However this theorem does apply to the solvable subgroup GT of p x p lower
triangular matrices g of the form
9 =
(9u
0
V 0
0
322
92p
0 \ 0
Spp/
(5.31)
From Example 5.1.4 a maximal invariant in the sample space under GT is
R — ( i ? 2 , . . . , R p ) ' and the corresponding maximal invariant in the parametric
space is A = (62,...,6 P )'. From (5.29) R has a single distribution under
H IQ and H 2 o and it has a distribution which depends continuously on the pi-
dimension parameter Fix under Hi\ and on the p-dimension paramete r r 2 A
under H 2X , where
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104 Group Invariance in Statistical Inference
T IX = { A : tf< > 0, t = 2 , . . . , p i + l , Si=0,i = pi
PI+I
+ 2 , . . . , p , = pf = A } ,
;=2
r 2 > = { A : tfi = 0, t = 2 , . . . , p i + l , 6i>0, i — pi
+ 2 , . . . , p , g ft = 4 = A>. (5.32)
i= P 1 +2
Let
with tfi/p2, if p 2 > 0
0, if p 2 = 0 .
Because of the compactness of the reduced parameter space {0 } and r1A
and the continuity of fx,n( r ) in n we conclude from Wald (1950) that every
minimax test for the reduced problem in terms of the maximal invariant R is
Bayes. Thus any test based on R with constant power on TJX is minimax for
problem 7 if and only if it is Bayes. From (5.29) as A ~* 0 we get (for testing
H 10 against Hi\)
P I + I
j=2 , i > j
+ B ( r , A , i j ) (5.33)
where S ( r , A, 17) — o(A) uniformly in r and r>. It is obvious that the assumptionof Theorem 5.1.1 is satisfied with
P I + I
U= J^Rj = Ri, h(\) = b\
j=2
where b > 0. The set p 2 — 0 is a single point n = 0, so £rjj assigns measure one
to the point T? — 0. The set Fix is a convex p\-dimensional set wherein each
component r>i— o(h(\)). So for a Bayes solution any probability measure t\\,x
can be replaced by the degenerate measure £* A which assigns the probability
one tomeanri 0 - (J? 0,. . . ,J7°, , + 1 ) o f£ i* . Choos ing£ * A which assigns probability
one to the point whose j t h coordinate is = (N-j + l)~1
(N-j+2)-1
p21
(N-
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Some Mintmax Test in Multinormates 105
p + l){N-pi), j =2 , . . . , p i + l we get (5.6 ) from (5.33) . T h e condit ion (5.5)
follows from The or em 4.4.1. Hence we prove the following theorem.
T h e o r e m 5 .1 .5 . For Problem 7 the likelihood ratio test of Hit defined in
(4.38), is locally minimax against the alternatives H \ \ as X —> 0.
In problem 8 as X —> 0 (for testing H 2 Q against H 2 x)
A , „ ( r ) = l .V A- i + n + Y, r i ( S ' K + ( i V - i + 2 H.. 1 + 5 ( r , A , 7 i ) ,
/o,T,(r)
(5.34)
where B ( r , A, JJ) — o(/i(A)) uniformly in r, TJ. Since the set I ^ A is a p2-dimen-
sional Euclidean set wherein each component r/, — o(h(X)), using the same
argument as in problem 7 we can write
/ I
h,n( r )^o,x(dv)
- 1 + f, + YI r>(£ w + ( w - 3 + 2)%
j = p i + i
JVA= 1 + - i + f,+ 2 ^(Sf+c^-j'+iM + B ( r , A )
(5.35)
where B ( r , A ) = o(/i (A) ) uniformly in r and £ I A assigns measures one to
« + 2 < ) ' •
Let iifc be the rejection region, given by,
Rk = {x: U{x) = n + kf 2 > c a } (5.36)
where k is chosen such that (5.36) is reduced to yield (5.5) and c Q depends on
the level of significance a of the test for the chosen k. Now choosing
o (N-p + 2)(N-p+l) „
p ( N - p + 2)p 2
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106 Group Invariance in Statistical Inference
we conclude that the test <p* with the rejection region
B* = U(x) = ft + ^ — - ' 2 > c a jwith PO ,A (W* ) = Q satisfies (5.6) as A -» 0. Moreover any region R k of the
form (5.36) must have k = in order to satisfy (5.36) for some From
(4.37) it is easy to conclude that the test 4>' is L B I for problem 8 as A—» 0. T h e
iE 2-test which rejects H20 whenever r 2 — f i +f 2 > c a does not coincide with 0*
and is locally worse. From Sec. 4.3 it is evident that the power function of R?
test depends only on p 2 and has positive derivatives everywhere in particu lar
at p 2 = 0. From (5.5) with R — R*, h{\) > 0. So we get the following theorem.
T h e o r e m 5.1.6. For Problem 8 the LBI test <p* which rejects H 2 a whenever
f 1 + N ~f l f 2 > c a where c a depends on the size a of the test, is locally minimax
against the alternatives as A —* 0.
R e m a r k s . In order for an invariant test of H 2 n against H 2 x to be minimax,
it has to be minimax among all invariant tests also. However, since for an
invariant test the power function is constant on each contour p 2 = p\ = A,
"minimax" simply means "most powerful". T h e rejection region of the most
powerful test is obtained from (5.34), from which it is easy to conclude that
[/A,7j('")//o,n(r)] depends non-trivia lly on A so that no test can be most powerful
for every value of a.
5 .2 . A s y m p t o t i c a l l y M i n i m a x T e s t s
We treat here the setting of (5.1) when A — > 00. Expressions like o(l),
o(H(\)) are to be interpreted as A - * 00. We shall be concerned here in maximizing a probability of error which tends to zero. T h e reader, familiar with
the large sample theory, may recall that in this setting it is difficult to com
pare direc tly approximations to such sm al l probabilities for different families
of tests and one instead compares their logar ithms. Whi le our considerations
are asymptotic in a sense not involving sample sizes, we encounter the same
difficulty which accounts for the form (5.39) below.
Assume that the region
R={x:U(x)>c a } (5.37)
satisfies in place of (5.5)
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Some Minimal Test in Mu Kin urinates 107
= (5.38) P x>n(R) = 1 - e x p { - i r ( A ) (1 + o( l) ) }
where #(A) —i co as A —» co and o(l) term is uniform in TJ. Suppose that
p(x,A,7j)^ l i A (d7i)
/p{x,0 ,7j )^ 0 , A (dj ; )
= exp{ff(A)[G(A) + R(\)U{x)} + B(af ,A)} (5.39)
where sup^ |B(a: ,A) | = o{H(X)), and 0 < cj < R(X) < c 2 < oo. One other
regularity assumption is that c Q is a point of increase from the left of the
distribution function of U when 6 — 0 uniformly in n, i.e.
i n fP o „ (E / > c a - e ) > a (5.40)
for every e > 0.
T h e o r e m 5.2.1. / / U satisfies (5.38)-(5.40) and for sufficiently large A
there exist £ 0 , A and £ i ( A satisfying (5.39), then the rejection region R is asymp
totically logarithmically minimax of level a for testing Ho -6 = 0 against the
alternatives Hi : 6 — A as A -» oo, i.e.,
l i m i n f , [ - l o g ( l - P A , , { * } ) ] = 1
A - » sup^ A e Q a inf ,[- log(l - P a , , ( 0 A rejects /Jo))]
Proof . Assume that (5.41) does not hold. Then there exists an e > 0 and
an unbounded sequence V of values A with corresponding tests d>\ in Q a whose
critical region satisfies
P X ,V {R} > 1 - e x p { - # (A) (1 + 5e)} (5.42)
for all IJ . There are two possible cases, (5.43) and (5.46). If A € T and
- 1 - ( 7 ( A) < R(X) c a + 2e, (5.43)
consider the a priori distribution £A (see Theorem 5.1.1) given by and r A
satisfying
r A / ( l - n ) = exp {if (A) (1 + 4e)} . (5.44)
Using (5.42) and (5.44) the integrated risk of any Bayes procedure £f A
mustsatisfy
rl(Bx) < rl(M < (1 - r x)a + r A e x p { - J / ( A ) (1 + 5e)}
- (1 - Tx)\a + e x p { - e H ( A ) } ] . (5.45)
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Some Minimal Test in Multinormo/es 109
with Sup \B(r,T), A)I — o(l) as A —• 0 0 . From (4.5) putting n p = 1, 7)i — 0,
i < p, in (5.49) the density of U being independent of n we see that
P X „{U < c a } = e x p { ( C a - 1)[1 + o ( l ) ] } (5.50)
as A —• oo. Hence (5.38) is satisfied with H(X) = | ( 1 — c a ). Now letting
£ i i A assign measure one to the point Vi = " — Vp-i — 0 5 Vp — 1 A N A £O,A
assign measure one to (0,0) we get (5.39). Finally (5.40) is trivial. From
Theorem 5.2.1 we get Theorem 5.2.2.
T h e o r e m 5.2.2. For every p,N,a, Hotelling's T 7 test is asymptotically
minimax for testing HQ .6 — 0 against H\ : 6 — A as A —> co.
From Sec. (5.1) we conclude that no critical region of the form £ ai R{ > cother than Hotelling's would have been locally minimax, many regions of this
form are asymptotically minimax.
T h e o r e m 5.2 .3 . c < 1 and 1 — a x < a 2 < •• • < a p , then the critical region
{ £ tti Ri > c} is asymptotically minimax among tests of same size as A —> 0 0 .
Pr oo f. Th e maximum of £ V f j 2~2i>j Vi subject to £ ; Ojr< < c is achieved
at r\ — c, T2 — • • • — r p — 0. Hence the integration />,,()") over a smal l
region near that point yields (5.50) wi th c„ replaced by c. Since a },'s are
nondecreasing in j it follows from (5.49) that £I ,A can be chosen to yield (5.39)
with U — Yl{aiRi. Again (5.40) is trivial. •
E x a m p l e 5. 2. 2. Consider again Problem 2 of Chapt er 4. From E x a m
ple 5.2.1 with p = pi, R = (Ru ..., R Pl )', r, = (n,,... ,n Pl )' we get
= e x P S ~ -^E^E* ( l + B(r,A,>))) (5.51)
with sup,.^ \B(r,\,n)\ — o(l ) as A -> 00. Using the same argument of the
above theorem we now prove the following.
T h e o r e m 5.2.3. For every p i , N, a the UMPI test is asymptotically min
imax for testing HQ : 61 — 0 against Hi :6 = \as\—* 0 0 .
E x a m p l e 5.2 .3 . In Problem 3 of Ch apter 4 we want to find the asy mp
totically minimax test of H Q : 6~i — 62 — 0 against the alternatives H x : 61 — 0,
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110 Croup Invariance in Statistical Inference
62 = A as A —» co. In the notations of Ex amp le s 5.1.4 and 5.2.1 we get,
r
i - E 1 1 r J> f i +*a = E V + P 2 T v T = (n, -,Tn+toY* v = ( * h , • • • . * , + » ) ' .
ni = 0, i = 1,. . . , pi + P 2 under H 0 and rj, = 0, i = 1 ,. . , ,p i under H\. Hence
/o,n(r)
PI+ PJ
- l + f i + £ r i £ » K
where sup,.,, \B(r,A,n)| = o (l ) as A - t 00 . From (4.9) when 5a
61 = 0 we get
= A
(5.52)
00,
/ ( r , , f 2 | A )
/Cl.falO)= exp j^ [ - l + f i + f 2 ] ( l + B ( f i , f 2 . A ) ) j (5.53)
where sup, .^ \B(f t ,f 2 ,\)\ = o(l) as A -* 0. Letting fa in (5.39) assign
measure one to the single point n — 0 we get from (5.52)
/ A , , ( rKi .x(di j)
- l + f , + £ r , 2 *>=Pi + l i > j
. - ? .A ) ) |
(5.54)
Let Rk be the rejection region of size a, based on R\,R 2 given by
R k = {x : U(x) = R t+kR 2> c a } (5.55)
where k is chosen such that (5.54) is reduced to yield (5.39) and R k for the
chosen k satisfies (5.38) and (5.40). Now letting 1/1 = • • • = T ) P l + p l _ i = 0,
- 7 P l + P ) = 1 we see that (5.54) is reduced to yie ld (5.39) with U{x) = Ri + R 2 .
From (5.53)
P(U(x) <c'-)= exp j fa - 1)(1 + o ( l ) ) j . (5.56)
Hence Hotelling's test which rejects fi0 for large values of R\ + R 2 satisfies
(5.38) with H(X) = f (1 - c' a ). The fact that Hotelling's test satisfies (3.39) is
trivial . Since the coefficient of r p i + i in the expression inside the brackets in the
exponent of (5.54) is one, any rejection region R k must have k = 1 to satisfy
(5.39) for some £ \ t \ . From Theorem 5.2.1 a region R k with k / 1 and which
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Some Minimax Teat in Multinormales 111
satisfies (5.38) and (5.40) cannot be minima x as A —• co. Us ing Th eo re m 4.1.5
we now prove the following:
T h e o r e m 5.2.4. Hotelling's test which is asymptotically best invariant is
asymptotically minimax for testing HQ I fli = &2 = 0 against H± : 6 T —
0,
oj = A as A —* oo.
R e m a r k s . From Theorem 5.2.3 it is obvious that there are other asymp
totically minimax tests, not of the form Rt, for this problem. It is easy to see
that
P Xin{R1 + (l-c)-
l
R 2
> 1) = l - e xP
{ - ( l o g A
2
- l o g ( ( l - c ) ( l + c ( l ) ) ) } ,
P a
' " ( A i + ^ r 1 * 2 - C o ) =1 ~ e x p { ~ ~C o ) ( 1 + 0 ( 1 ) )
Thus the fact that the likelihood ratio test which rejects HQ whenever1~/l*fi1
ft *' 5 c and the locally minimax test satisfy (5.40) and (5.38) is obvious
in both cases. But from Theorem 5.2.4 these two tests are not assymptotically
minimax for this problem.
5 .3 . M i n i m a x T e s t s
In this section we discuss the genuine minimax charac ter of Hotelling's T 2
and the test based on the square of the sample multiple correlation coefficient
R 2
. These problems have remained elusive even in the simplest case of p — 2 or
3 dimensions. In the case of Hotelling's T 2 test Semika (1940) proved the U M P
character of T 2
test among all level a tests with power functions depending
only on & — Np'H~1 p. In 1956 Ste in proved the admissib ility chara cter of T 2
test by a method which could not be used to prove the admissibi lity character
of R 2 - tes t . Gi r i , Kiefer and Stein (1964a) attacked the problem of minimax
property of Hotelling's T 2 -test among all level a tests and proved its mini
max propery for the very special case of p — 2, N — 3 by solving a Fredhol m
integral equation of first kind which is transformed into an "overdetermined"
linear differential equation of first order. Linnik, Pliss and Salaeveski (1969)
extended this result to N — 4 and p — 2 using a slightly more compl ica ted
argument to construct an overdetermined boundary problem with linear dif
ferential operator. Later Salavesk i (1969) extended this result to the general
case of p = 2.
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112 Group /nvuriaiice in Statistical Inference
5.3.1. Hotelling's T 2 test
In the sett ing of Exa mp le s 5.1.1 and 5.2.1 we give first the details of
the proof of the minimax property of Hotelling's T 2 test as developed by
Giri , Kiefer and Stein (1964a), then give the method of proof by Lunnik,
Pliss and Salaeveski (1966). In this setting a maximal invariant under G j f p )
is R = (R1 ,... } R P )' with = NX'(S + NXXy^X and the corre
sponding max imal invariant in the parametr ic space is A = (6i,..., S p )' with
£ p St = N p ' £ - 1 u = S. From (5.18) R has a single distribution under H 0 and
its distribution under Hi : 6 = X (fixed) depends continuously on a p - 1
dimensional parameter
T = | A = (6 1 6 P )': « 5 , > 0 , =
Thus there is no U M P I test under Gr(p) as it was under Gt{p). Let us write
/ A ( r ) as the pdf of as given in (5.18) . Because of the compactness of the
reduced parametric spaces {0} and T and the continuity of f&( r ) i n A we
conclude from Wald (1950) that every minimax test for the reduced problem
in terms of R is Bayes. In particular Hotelling's T 2 test which rejects He,
whenever U = £ J R\ > c, which is also G r - i n variant and has a constant power
function on each contour Np'T,~1 p = X, maximizes the minimum power over
H\ if and only if there is a probability measure £ on T such that for some
constant Ci
' r / o t » \m 1 = ] *t (5.57)
according as
<
= > c
1 K < J
except possibly for a set of measure zero where c depends on the specified level
of significance a and ci may depend on c and the specified X. An examination
of the integrand of (5.57) allows us to replace it by its equivalent
« d A ) = c, if Y > - e . (5.58)
r Jo\ r
) ,
Obviously (5.57) implies (5.58). On the other hand, if there are a £ and a
constant C[ for which (5.58) is satisfied and if r* = ( r j , . . . , r ' ) ' is such that
£ i r j = c' > c, then writing / = ^ and r" = ft* we conclude that
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Some Minimax Test in Multinormales 113
because of the form of / an d the fact that c'/c > 1 and £ p r** = f E i r t = c-
This and similar argument for the case c' < c show that (5.58) implies (5.57).
Of course we do not assert that the left-hand side of (5.58) still depends only
on 2%n if £ p n / c.
The computation is somewhat simplified by the fact that for fixed c and
A we can at this point compute the unique value of ci for which (5.58) can
possibly be satisfied.
Let R = (Ri,..., Rp-i}' and let f&{f/u) be the conditional pdf of R given
£ p = i Ri — it wi th respect to the Lebesgue measure , which is continuous in
f and u with > 0 and1 r
« < it < 1. Denote by / " ( u ) the pdf of U = E i which is continuous for 0 < u < 1 and vanishes elsewhere a nd
which depends on A only through 8. Then (5.58) can be written as
j /A t fMtfdA) =1 /;*<<=)
fo(f\c) (5.59)
if Ti > 0, E i_ 1 T
i< c
- The fian
^ integral in (5.59), being a probabilitymixture of probability densities is itself a probability density in f as is fo(f\c).
Hence the expression inside square brakets of (5.59) is equal to one. From (4.5)
r ( f ) r ( V ) i 2 '
. C ^ ) ( a _ c ) ^ - P - , 0 ^ , E ; | ) . { 5 . 6 0 )
Using (5.60) we rewrite (5.58) as
J r I 1 i>3 > >=1
J V - i + 1 1 r $ - \ * , . A * , / J V /j cA
for all r with rj > 0, E i ri = c - Write T i for the unit (p - l)- sim plex
2 '2' 2 i M ' ' 1 2 ' ! ' 2
(5.61)
r ! = { ( A , . . . , / 3 p ) : f t > o , X > = i }
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114 Group Invariance in Statistical Inference
where /3; — 6i / X. Let U — 7 — cA and £* be the probabili ty measure on
T i associated with £ on T. Since — t]{6A), (5.61) reduces to
J r
> I )=1 i>j i fe=lv
'
= * ( f , f ; i , ) ( - >
for all ( i i , . . . , f p ) with ' i = 7 a i m ii > 0 and hence, by analyticity for all
(h t P ) with gjfft = 7-
From (5.62) it is evident that such £*, if it exists, depends on c and A only
through their product 7 . Note that for p — 1, T i is a single point but the
dependence on 7 in other cases is genuine.
C a s e p = 2, N = 3 . Sol ut io n of G i r i , K i e f e r a n d S t e i n
Since 0 ( | , — (1 + x)exp(^x), from (5.62) we obtain
jf [i + ( 7 - - ft)]<TO) = ^ " ^ ( f , i ; |-
(5.63)
with fi = 7 - f 2 , A = 1 - / 3 2 . We could now try to solve (5.63) for by using
1 . 1 .21 2'the theory of Meijer transform with kernal ^(1 ,4; hx). Instead we expand both
sides of (5.63) as power series in f 2. Let
Jo
be the ith moment of /3 2. From (5.63) we obtain
(a) l+t-*m=B,
(b) -(2r - 1) p r _ ! + (2r + 7 W - 7^+1 = BT ( - - ) r ( i ) J v
where B = e~^ y d>(^, 1; £ 7 ) . We could now try to show that the sequence {p,}
given by (5.64) satisfies the classical necessary and sufficient condition for it
to be the moment sequence of a probability measure on [0,1] or, equivalently,that the Laplace transform
G O
3= 0
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Some Minimal Test in Muttinormales 115
is completely monotone on [0, oo), but we have been unable to proceed suc
cessfully in this way Instead, we shall obtain a function m1(x) which we
prove below to be the Lebesgue density dt]*{x)jdx of an absolutely continuous
probability measure f* satisfying (5.64) and hence (5.63).
The generating function
C O
of the sequence {fa} satisfies a differential equation which is obtained by mul
tiplying (5.64) (b) by f _ 1 and summing from 1 to oo:
2 t 2 ( l - t ) V J ' ( t ) - ( f 2 - 7 t + 7 M t )
= B t [ ( l - t r 1 / 2 - l ] + 7 [ t ( l - / 0 - l ]
= S ( ( l - t ) " 1 / 2 - ( - 7 . (5.65)
This is solved by treatment of the corresponding homogeneous equation
and by variation of parameter to yield
mf j M zl 2 i
B
] d T
n ' ( l - i ) l / a
J 0 [3T#-Ift* 2 T 2 ( l - T ) 1 / 2 2 T ( 1 - T ) J '
(5.66)
the integration being understood to start from the origin along the negative
real axis of the complex plane. The constant of integration has been chosen
to make VJ continuous at 0 with VJ (0) = I , and (5.66) defines a single-valued
function on the complex plane minus a cut along the real axis from 1 to oo.The analyticity of ip on this region can easily be demonstrated by considering
the integral of ip on a closed curve about 0 avoiding 0 and the cut, making
the inversion w = | , shrinking the path down to the cut 0 < w < 1 and using
(5.67) below. Now, if there existed an absolutely continuous £* whose sui tably
regular derivative m 7 satisfied
j m7
(a; ) / ( l - tx)dx = i>(t), (5.67)Jo
we could obtain m y by using the inversion formula
m^(x) = ( 2 i r i x ) _ 1 l i m t y ( x - 1 + it) - T / > (X - 1 - ie)]. (5.68)
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116 Group Invariance in Statistical Inference
However, there is nothing in the theory of the Stieltjes transform which tells us
that an m y(x) satisfying (5.68) does satisfy (5.67) and, hence (5.63), so we use
(5.68) as a formal device to obtain an m 7 which we shall then prove satisfied
(5.63).
From (5.66) and (5.68) we obtain, for 0 < x < 1,
m-,(x) —2rr x^
2
(l - x)1
' 2
\ J 0 | l + u (1 + u) 3
' 2
B v>l 2
+ B
Ja 1 - « J
(5.69)
In order to prove that a%*{x) — m 7 ( x ) dx satisfies (5.63) with £" a probability
measure we must prove that
(a) m T ( i ) > 0 for almost all x, 0 <x < 1 ,
(b) f 01
m1{x)dx = l (5.70)
(c) fii = xm-f(x) dx satisfies (5.64) (a)
(d) p r — j 0 xT
m1{x) dx satisfies (5.64) (b) for all r > 1.
The first condition follows from (5.69) and the fact that B > 1 and u 3 / 2 ( l +
u) < (1 + w) 3-' 2 for u > 0. The condition (d) follows from the fact that m 7 ( x )
as defined in (5.69) satisfies the differential equation
w! (x) + m 7 ( i ) | + ( l - 2 x ) / 2 x ( l - x ) = B/2irx1/2(l-x) 3< 2 , (5.71)
so that an integration by parts yields, for T > 1,
( r + l K - r ^ r _ , = / [(r + l ) x r - rx r
-l
\m y(x) dx
Jo
= / dx Jo
= uT (l - 7 / 2 ) + 7 iiT+l /2 - « r _ i / 2
+ B r ^ + ^ / 2 T r ^ r !
which is (5.64) (b).
To prove (5.70) (b) and (e) we need the following identities involving con
fluent hypergeometric function <p(a,b;x). The materials presented here can
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Some Minimax Test in Multinormates 117
be found, for example, in Erde ly i (1953), Chapter 6 or Stater (1960). T h e
confluent hypergeometric function has the following integral representation
0 K 6; x) = T(a + j)T(b)/r{a)T(b + j) j \ JS>
5=0
when 6 > a > 0. T h e associated solution ip to the hypergeometric equation
has the representation
if a > 0. We shall use the fact the general definition of ip, as used in what
follows when a — 0, satisfies
iP(0,b;x) = 1. (5.74)
The functions T/J and <f> satisfy the following differential properties, identities
and integrals.
~<t>(a,b;x)= (j-l)b+l;x) + 0(o ,6 ; i ) , (5 .75 )
^ / > ( a , 6; x ) - $ ( a , 6; x) - ^ ( a , 6 + 1; x), (5.76)
^ ( a , b ; x ) = i - ' o K a - H l W a + l . i ; ! ) - ^ , ! ; ! ) ] , (5.77)
(a - b + l)0(o, 6; x) - a0(a + 1,6; x) + (b - l ) 0 ( a , b~l;x) = 0, (5.78)
60(a ,6 ; i ) - bd>(a - 1 , 6 ; a ; ) - x 0 ( o , 6 + l ; i ) = 0 , (5.79)
ih(a, 6; x) - ai/>(a + 1, b; x) - ip{a, b - 1; x) = 0, (5.80)
( 6 - a K > ( a , 6 ; x ) -xih(a,b + l;ar) + ^ - ( o - l , fc;«) = 0 , (5.81)
rJo
e~ x(x + y)-l <f>[ - , l ; i <tt = * r , l ; y for y > 0 ,
(5.82)
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118 Group Invariance in Statistical Inference
J g e-*(s + y)-vQ,l;z)<te
(5.83)
Using (5.73) , in terms of hypergeometric functions, for 0 < x < 1, the m 7 ,
defined in (5.69), can be written as
= h [ , '^ ./.{^ / '<l- 1 ; ^)*' 1 - l i T / 2 »- r (l)*(i l i ^)+ </>(^,1 ,7/2j jf v - l e - < " t 2 d v - v ^ e ^ ^ d v J
° r r ? r „ . ( e " ^ f i i : T / 2 ) ^ i , i ; 7 ( i - » ) / 2 )2TT[X(1 - ( \ 2 /
E/2r(|)v-(|,l;7/2)}. (5.84)
We now prove (5.70) (b) to establish that m-,(x) is an honest probability
density function. From (5.73), (5.72) and (5.82) we get
/ VTTt—^*(i.i;7(i-»)/?)#Jo 2ir x f l - i ) J
= f VT7T~ v r ^ ( l , 1 ; 7 * / 2 ) tteJo 2ir[x(l - l ) ]a
7o 2 , r [ * ( l - * ) ] » 7o 1 J
dt
(5.85)
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Some Minimal Test in Mull in orm ales 119
From (5.85), (5.84) and (5.72) we get
„i r i4 e z f
r(|) h dx
= 2 ^ , l ; ^ ( i , l ; , ) - 0 ( I , l ; ^ ( ^ , l ; S
where 2z = 7 .
We now show that
tf'(z)-#(*)=0, (5.86)
from which it follows that
H{z)=cel
for some constant c.
B y direct evaluation in terms of elementary integrals when 7 — 0 we get
(using (5.85))
/ m a(x) dx = 1; Jo
hence, (5.70) (b) follows from (5.S6). To prove (5.86), we use (5.75) and (5.76),
which yield
H\z)-H(z) = <b{^,2 ] z^i>^\-,z^ + # / | , l ; » \ ^ Q 1 i 5 ^
-2Kii;^G'2;s)+^G'2;^G ,M)1 1
To this expression we now add the following four left-hand side expressions,each of which equals zero
1 1 . ^3i> U . 1 ; * ~ n H ^ 2 ; 2 - -<P[ - , 2 ; z + 0 - , l ; z
2 T \ 2
2* U
( N1 . / 3
2 " V 2
^ - , 2 ; * - - t f - , 2 ; * - t f - , l ; z1
- 0 1 2 - 2 ^1 . (Z
2 T \ 2
to obtain H' — H — 0, as desired.
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120 Group Invariance in Statistical Inference
We now verify (5.70) (c). Prom (5.72) and (5.79), with a = §, b = 1 we
obtain
1 (1+7?/) e™' 2
dy = M 7 -,l;1 /2 + 7 0 - , 2 ; 7 / 2 / 4
(5.88)
Using (5.70) (6), which we have just proved, and (5.85) and (5.88) we rewrite
(5.70) (c) as
1-1 AI
1 - |, mt/aJJ jf u + 7(1 - $}] M * ) *
( l + 7» )
2 T [ » ( 1 - V ) ] I / 21 ; 7 / 2 W M , ?
- e ^ r U 1 1 ; 7 / 2 )\dy
1
7 g ^ ( l , l ; 7g / 2 ) , , f l 0(1,1; Tg/g)
2 ^ ( 1 - f + 7 0 2 ^ ^ ( 1 - 2 / ) ] ! ^ »
r ( | ) f ( f . 1,7/2) y 1 ( l + 73 / ) e ^ 2
0 ( | , l ; 7 / 2 ) I 2ff [v(l - J/)]V2
= f 1 7 ^ ( 1 , 1 ; 7J//2)
Jo 2T[J /(1 — J f) ] 1 ' 2
dy
1_ A l „ / 3
+ 2 r ( - ] ^ ( - , l ; 7 / 2 ) - - r [ - ] ^ [ - , l ; 7 / 22 V2(5.89)
Using (5.81), with a = 1, 6 = 0 and (5 .72) , (5.73), (5.83) we get
f 1 7 g ^ ( l , l ; 7 » / 2 ) , _ f 1 M 0 , 0 ; 7 y / 2 ) - » ( l , O ; 7 y / 2 ) ] ,
/ „ 2 ^ ( 1 - , ) ] ^ ^ - y 0
d »
= 1 - f
1 _ dy
Jo * Jo[3 /(1 - » ) ] ^ a
= l - j T ( l + r V Q ^ 7 * / 2 ) e - ^
= 1 + r(|) ^(I^^A-^d-HT/s) / 2 .
(5.90)
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Some Afinimai Test in Mv.lttnorma.les 121
Thus (5.89) and (5.90) imply (5.31) and, hence (5.70) (c).
C a s e p = 2, N = 4. Solution of L i n n i k , Pl i s s and Salaevski
From (5.62) the problem consists of showing (with 0i — /J) on the interval
0 < 8 < 1 there exists a probability density function £{/3) for which
= 0 ( 2 , l ; 7 / 2 ) . (5.91)
With a view to solving this problem we now consider the equation
j * e
" < ' - « / J
^ 2 , | ; M / 2 ^ { | | i (7 - t l) (l - 8)/2)mW
= l 4l <\M ̂ -0)/2jmdf3. (5.92)
Using the fact that
we rewrite (5.92) as
1 6 - ^ 0 ( 2 , 1 ; ^ ) [ 1 + 7 -10 - d d - / 3 ) K ( W
- / 1
e - ^ / 2 [ 1 + 7 _ 7 ^ K ( A ) ( f / J ( 5 9 3 )
Jo
From (5.93) we get
/" e - ^ 2 [ ( r - 2r 2W~l + (1 + 7 + 7 r + 2r 2
)8 r
Jo
- (7 + irW +1
}t:(0)d0 - 0, for r = 1 , 2 , . . . . (5.94)
Let
r = l
be an arbitrary function represented by a series with radius of convergence
greater than one. Multip lying the rth equation in (5.94) by o r _ i and taking
the sum over r from 1 to 00 we get
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122 Group Invariance in Statistical Inference
f1
s:(2/33 - 2/3 2 )/" + (-5/3 + 60 2 - f& 2 + 7 / 3 3 ) / '
! - l + o 0 - ^ + f 0 2 ) ,
= J LUMP) d0 Jo
= 0 (5.95)
where L{f) denotes the differential form inside the square brakets. Applying
the green formula to L(f) and choosing a small e > 0 we get
where the adjoint form L*(£) is given by
L * ( 0 = 20 2(0 - 1)C + 0{-Z + 60 + 7 /3 - 7 /3 2 K'
and the bilinear form L[f, £] is given by
m% =2
^2
( ! - ^ ) ( ^ - / ' o - \0+7^2(i - mi n •
Using the thoery of Frobenius (see Ince, 1926} we conclude that
C C C O
(5.96)
(5.97)
(5.98)
(5.99)
£=0
with tto / 0, Co / 0; are two fundamental solution of L*(Q — 0 on the interval
(0,1)- Similarly
£ n = £ 96(1 - /J)f c , j i a - (1 - /3)"5 £ fefcd - 0)1
(5.100)
k=0 k=0
with 3o ^ 0, fto # 0 are also two fundamental solutions of the same equation
on the interval (0,1). Consequently any arbitrary solution of the equation
L*(£) — 0 is integrable on [0,1] and we assert that there also exists a solution
of the equation L*(£) — 0 for which
h m L [ M ] | ; - < = 0. (5.101)
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Some Minimax Test in Mullinormales 123
It can also be shown that any linear combination of the solutions in (5 .99)
(similarly for solutions in (5.100)) satisfies (5.101) provided that £(• ) = £ i 2 ( - ) .
Thus £ - £12 is a solution of (5.95) and hence of (5.91).
In the following paragra ph we show that £ i 2 does not vanish in the interval
(0,1). Write
0 = 1 - * , * ( a r ) = A 5 * y f & i ( l - 4 - (5.102)
The equation L"(t\) — 0 becomes
2x{\ - x)8" + ( - 1 + 4x--,x + -yx 2 )S' + ^(1 + jx)$ = 0 . (5.103)z
Substituting u(x) = 6'(x)j8(x) we obtain the Riccati equation
l - 4 s + 7 a - 7 s ' _ 3 + ^ 7 *
2x{l-x) 4 x ( l - x ) v '
Let us recall that 8(x) — E t ^ o ^ ) 1 * ' • > 1*1 < !• Ass um e that 9{x) vanishes
at some point in the interval (0,1). Since 0(0) = 1, among the roots of the
equation ${x) = 0 there will be a least root. L et us denote it by XQ. Then
the function 0(x) is analytic on the circle \x\ < x$. From (5.103) we get
£?'(0) — | which implies that u(0) = —| . Hence u(x) cannot vanish in the
interval (0, Xrj), since at any point where it vanishes we would necessarily have
u'(x) > 0 whereas (5.104) shows that at this point u'(x) < 0. T he fact that
the function u(x) has a negative value implies that it decreases unboundedly
as we approach the point XQ from the left (u' / 0 since which in turn
contradicts (5.104). As a result £ = £ i 2 does not vanish in the interval (0,1).
L e t us now return to (5.91), which, as we have seen, is satisfied by £ — £ 1 2 .
By the method of Laplace transforms it is easy to obtain the relation
/ x b
-l
{l - x y - ^ e i *tt>(a,b;px) <p(a - b,c - b; q(l - x))Jo
r(t)r(c - b)
r(c)-4>(a,c;p + q), 0<b<c. (5.105)
Letting t\ = 7 T , and multi ply ing both sides of (5.91 ) by r 5 (1 —r ) ' and
integrating with respect to r from 0 to 1 we obtain with a = 2, 6 = | , c = 1,p = 7£72,? = 7 ( l - W 2 ,
| y 2 , l ; ^ W = p ( H ; ^ ) ? ( W . (5.106)
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124 Group Invariance in Statistical Inference
Now to obtain (5.91) it is sufficient to make use of the homogeneity of the
above relation and to normalize the function £12.
5.3.2. R?-test
Consider the setting of Example 5.1.5 for the problem of testing H 0 : p 2
— 0
against the alternatives Hi : p 2 = A > 0. We have shown in Chapter 4
that among all tests based on sufficient statistic (X,S) of HQ against the
alternatives p 2
> 0 the R 2
-test is UM P invariant under the affine group ( G , T )
of transformations (9,t) , g e G , t e T operating as (X,S) —• (gX + t, gSg')
where g is a nousingular matrix of order p of the form
a = (%m 0
with 3(H) of order 1 and t is a p-vector.
For p > 2 this result does not imply the minimax property of R 2
-test
as the group ( G , T) does not satisfy the condition of the Hunt-Stein theo
rem. As discussed in Example 5.1.5 we assume the mean to be zero and
consider only the subgroup Gr ( p ) for invariance. This subgroup satisfies the
conditions of the Hunt-Stein theorem. A maximal invariant under GT-(P)
is R = (Rg,... ,Rp)' with a single distribution under Ho but with a dis
tribution which depends continuously on a (p — 2)-dimensional parameter
A — (6 2 , • • • ,S PY with £ J = A 6 j — p 2 — A under St- The Lebesgue density
function f\,7){ r
) °^ ̂ under Hi is given in (5.29). Because of the compactness
of the reduced parameter spaces {0} and
r = 6 p y, 6i>o, £ 4 = A ] (5.107)
and the continuity of fx,v{ r ) m 1" '* follows that every minimax test for the
reduced problem in terms of R is Bayes. In particular, the Jf 2-test which has
a constant power on the contour p 2 = A and which is also GT (p) invariant,
maximizes the minimum power over H i if and only if there is a probability
measure on Y such that for some constant cj
according as
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Some Minimax Test in Multinormales 125
except possibly for a set of measure zero. An examination of the integrand in
(5.108} allows us to replace it by the equivalent
/|McktfA3««i i f X > , = c- (5-109)
Obviously (5.108) implies (5.109). On the other hand if there is a £ and a
constant c\ for wh ich (5.109) is satisfied and if f = (r~2>---.*p)' s u c n t n a t
Y % f i = c 7 > c, then wr it ing / = and r' = cf/c' we conclude that
/ ( f ) = / ( c V / c ) > / ( r - ) = C l
because of the form of / and the fact that c'/c > 1 and ^ j j r * = c - Note that
in (5.29) 7 i "1 ( l — A) — 1 — - J2j>i 6 jfai a n d t h a t ti > °- T n i s ^ similar
argument for the case c' < c show that (5.108) implies (5.109). T h e remaining
computations in this section is somewhat simplified by the fact that for fixed
c and A we can at this point compute the unique value of C\ for which (5.109)
can possibly be satisfied.
L et R = (R2, ..., Rp-i)' and write /x,Tj(r|i/) for the version of conditional
Lebesgue density of R given that E 2 Ri = U = u which is continuous in f and
for Ti > 0 and E i _ 1 r« < w < 1 and zero elsewhere. Also write (u ) for the
Lebesgue density of R 2
— E 2 Ri = U which is continuous for 0 < u < 1 and
vanishes elsewhere and depends on A only through S - £ 2 ^ , - Then (5.109)
can be written as
/ / A,,(r|c)£(dA) - c i /c*(0
75(c)J
for fi > 0 and E S P 1 r> < c - The integral in (5.110) , being a probability mix tur e
of probability densities, is itself a probability density in f , as is /o o (r| c ). Hence
the expression in square brackets equals one. From (4.35) with 0 < c < 1
= q - A ) ^ - » r ( ^ - i » , ( p _ 3 ) _ i { N _ p _ 2 )
M ) r ( ( j v - P ) / 2 ) r ( ( p - i ) / 2 ) 1 '
\(N-l), 1 ( J V - 1 ) ; ± ( P - 1 ) ; C A ) (5.111)
where F(a, 6; c; x ) is the ordinary 2̂ 1 hypergeometric series given by
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126 Group Invariance in Statistical Inference
F(a,b;c;x) = 2^ - -- —*
r=0r ( « ) r ( 6 ) r ( c + r )
E(g ) r (fc)r _ r
. ( c ) r
(5.112)
where we write (a) r = T ( a + r)/T(a). From (5.110) the value of a which
satisfies (5.109) is given by
( i - A ) t t " - ' > r < i ( j y - i ) ) , , „ _ 3 , . ( N - P _ „
C l " r ( | ( / v - P ) ) r ( i ( p - i ) ) 1 J
FQ(JV- 1), i ( J V - 1); | ( p - 1);
CA). (5.113)
From (5.113) and (5.29) the condition (5.109) becomes
£ [ i + £ r i ( i - A ) / 7 i - t )
03=o 0 p=a r c i ( N - U )
* r ( i ( J v - j + l ) + ft-) r 4 r j ( l - A ) / 7 j ( l + 7r-') if t
i l r ( | ( i v - i + i))(2/3 j )! U + £ ^ ( ( 1 - * ) / • > ; - i ) J " ;
= F Q ( / V - 1 ) , l -(N-l); i ( p - l ) ; c A ) (5.114)
for all T with r, > 0 and £ J ri = c.
C a s e p — 3, N — 4 ( N — 3 if me an kn ow n) . Soluti on of G i r i and Kiefer
In this case (5.113) can be written as
. + 2n) >*3 *3
V(l-4)(i-^A)
r a g a ( l - A ) ( 2 n + l ) ( 2 n + 3)
( l - « a ) ( l - r a A ) l+
Ta 83
( l - f c ) ( l - r 2 A )
(5.115)
One could presumably solve for £ by using the theory Meijer transforms
with kernal F ( § , §; \\x). Instead they proceeded as for Hotel ling's T 2 problem,
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128 Group Invariance in Statistical Inference
measure on [0,1], or, equivalently that the Laplace transform £ ™ ltj(-t) slj\ is
completely monotone on [0, oo}. Unfortunately we have been unable to proceed
successfully in this way. We now obtain a function m t{x), which we then prove
to be the Lebesgue density d £ * ( x ) /d x of an absolutely continuous probability
measure C satisfying (5.118) and hence (5.115). That proof does not rely on
the somewhat heuristic development which follows, but we nevertheless sketch
that development to give an idea of where the 111,(1) of (5.123) comes from.
T h e generating function
C C
3=0
of the sequence {fij} satisfies a differential equation, which is obtained from(5.118) by multiplying it by t
n
~l and summing with respect to n from 1 to 00,
2(1 - t)(t - z)<p'{t) - t~l {t 2
- 2zt + z)<f>(t)
= tf,(l-t)-i - 1 - z f " 1 . (5.119)
Solving (5.119) by treatment of the corresponding homogeneous equation and
by variation of parameter, we get
m=W=T)\ k U l - r ) ( r -
+1
C r - * ) l ( 1 - * ) * 2 [ T ( 1 - T ) ( T - Z ) ] 1d r .
(5.120)
T h e constant of integration has been chosen to make <p(t) continuous at t = 0
with 0( 0) — 1 and (5.120) defines a single valued analytic function on thecomplex plane cut from 0 to z and from 1 to 00. I f there did exist an absolutely
continuous £* whose suitably regular derivative m z satisfied
L
1
m f (x) dx = 0( f ) ,
lo (1— tx)
we could obtain m c by using the simple inversion formula
m z(x) = - . lim27TIX ElO
<p(x^ +i e ) - 0 ( x - 1 - ie)
(5.121)
(5.122)
Since there is nothing in the theory of Stieltjes transforms which tells us that
an m z satisfying (5.122) does satisfy (5.121), we use (5.122) a formal device to
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Some Minimax Test in Multinormales 129
obtain m E which we shall prove satisfies (5.115). From (5.120) and (5.122) we
obtain, for 0 < x < 1,
( 1 - * * ) * f n f X
du« • ( « ) = ( 1 ; M ) , k /
- 2
(1 + u)(z + u p [u(l + u) (z + i*)]*
(1 + 1 i ) i ( z + «)
2jr(a:(l - as))i { B , Q , ( z ) + c , } (s ay). (5.123)
We can evaluate c c by making the change of variables V = (1 + u) 1 and using
(5.126) below.
We obtain
and
Q , - 2 ( l - z } - 1 [ l - ( l - ^ ) - ^
' ( l - z ^ i + f l - z ) ^ l - ( l - z ) i+ ( l - z ) " t log
. ( 1 - « ) * - ( ! - * ) * l + ( l - z ) * . '
Now to show that £*(da;) = m I (x) dx satisfies (5.115) with £* a probability
measure we must show that
(a) m , ( i ) > 0 for allmost all x, 0 < x < 1;
(b) fi m z(x) dx = l; (5.124)
(c) /ti = 1111,(1) da; satisfies (5.118a);
(d) u n = x n m,(a;) da; satisfies (5.118b) for TI > 1.
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130 Group Invariance in Statistical Inference
T h e first condition will follow from (5.123) and the positivity of B z and c,
for 0 < z < 1. The former is obvious. To prove the positivi ty of c z , we first
note that
this is seen by comparing the two power series, the coefficient of z j being
[ ( f j j / j f ] 3 and (j + 1), and the ratio of the former to the latter being n";=i(» +
I ) 2 / j { j + i ) > i. Thus we have B M > l - 2 . Substituting this lower bound into
the expression for c z and writing u = 1 — 2 the resulting lower bound for c z has
a power series in u (convergent for \u\ < 1) whose constant term is zero and
whose coefficent of v> for j > 1 is [(j + i ) " 1 - T 2
(j + ± ) / r ( j ) r ( j + +1)].
Using the logarithmic convexity of the T-function, i.e. V 2
(j+^) < !T(j)r(i + l ) ,
the coefficient of u> for j > 1 is > (j + | )- 1
- (j + l ) "1
> 0 Hence c z > 0 for
0 < z < 1.
To prove (5.124)(d) we note that m , ( i ) defined by (5.118) satisfies the
differential equation
1 - 2x + zx 2 1
m' x(x) + 7ro , ( i ) = B z /2nxHl - x)%{\ - zx), (5.125) x(l - x)(l — zx)
so that an integration by parts yields, n > 1
- z(n + 2)M-v+I + (1 + z)(n + l ) p« - n ^ n - i
= / {-z(n + 2)x n+1 +(1 + z)(n + l)x n -nx n~1 } m z{x) dx
Jo
= [ x n(l-(l + z)x + zx 2
)m' t(x)dx
Jo
= - ) - f i n _ l + 2 l l n - z u n + 1 + B t — ^ | ,
which is (5.118)(b).
T h e proofs of (5.124) (b) and (c) depends on the following identities in
volving hypergeometric functions F{a, b; c; x) which has the following integral
representation when Re(c) > Re(b) > 0:
F(a,b;c;x) = r ( & ) ^ _ 6 ) f - t ) M
( \ - tx)~*dt. (5.126)
We will also use the representation
2 ^ ( i l ; ^ ; x ) = 1 o g ( l ± f ) (5.127)
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Some Minimax Test in Multinormales 131
and the identities
F(a,b;c;x) = F(b,a;c;x),
( c - o - 1 ) F(a,b;c\x) +aF(a + l,b;c;x)
- ( c - 1) F{a,b;c - l;x) = 0,
lim [TicT'F^b-c^x)
= (a) + l ( f t U i ^ M A *-(„ + „ + i ) 6 + B + 1 ; „ + 2 ; x )
(JI + 1)!
(5.128)
(5.129)
(5.130)
for 7i = 0 , 1 , 2 , . . . ,
r ( l + A + n ) r ( l + y + /Q
= F \ l + A, ~ - v; 1 + X + - A, 1 + V;1 + J/ + ii; 1 - * j
+ F\ \ + X, \ - u; \ + X -t- fi\x \ f [ - i - A, ~ +1/; 1 + « + ./*;! — as
- j Q + A , | - i / ; l + A + Was) F ( | - A , | + + y + - ;£.j;
F(ffl ,6;c;x) = (1 - x ) C - " - 6 F ( c - a ,c - b;c; x). (5.131)
We now prove (5.124) (b) and (c). From (5.123), using (5.126) and (5.127) we
obtain
- ( I - 4 8 F ^ , i ; l ; ^ + ( l r / 2 ) | ; 1; 1 - ^
3 1
2 2
— F - , - ; 2; 1 - z] x F\ -1 1
2 2
(1 -— log
3 X 1 6 1 i 1
2ir(l - j ) t i i ( l - i ] l \ 1 — - z i ) - ?
l + (i-z)5(i-zx) I
(5.132)
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132 Group Invariance in Statistical Inference
The first expression in square brackets in ( 5 . 1 3 2 ) varishes, as is easily seen
from the power series of F in ( 5 . 1 1 1 ) . Using ( 5 . 1 2 7 ) , the integral in ( 5 . 1 3 2 }
can be written as
_ J y . (1 - zT f 1
dx
^ ( l " 3 ) ^ 2n + l Jo (l-x)i(l-zx)"
71=0 V
1 °° , m ( l \ r a oo°° z
m
( ^ ) m ^ (n + m ) ! < l - J t ) "2 * H ( m! ) 2 Z - B ! ( n + | )
m=0 v
' n=0v
3 '
m=0
r l - l
( i - f ) m +
= ( 1 _ z ) - 1 + l ( 1 _ z ) - i r ' _ — i i ^ _1
' 2V ;
; 0 ( i - 2 - o *
= ( 1 - z } "1
+ (TT /4) (1 - i r * * ( j , | ; 2 ; 1 - z ) . ( 5 . 1 3 3 )
From ( 5 . 1 3 2 ) and ( 5 . 1 3 3 ) we get
" 2 ' 2 " " / T V2'2| m , ( « ) ( f c = ( 7 r / 2 J F f - i , i ! l ; 4 r i F , r i i ; l
- P
1.1,*.-.)]
Hence, from ( 5 . 1 2 9 ) and ( 5 . 1 3 3 ) we obtain from ( 5 . 1 3 4 )
- F ( - ^ - ^ I ^ ) ] } . ( 5 . 1 3 5 )
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Some Minimax Test in Multinormales 133
Using (5.129) with a = b = c = 1 + e and (5.130) with n = 0 and
c = e —* 0, (5.132) reduces to
1 1 1 1
2 ' 2 ' mF[ - - , - ; l ; z ] F ( - , - ; 2 ; l - z ) + ( E / 2 ) F ( - , 2; 1-z )F[ - ; 2 ; *
3 1
2 ' 2 '
1 1
2 ' 2 '
(5.136)
Now, by (5.131), the expression (5.133) equals one if we have
- / • ( - i , 4 ; l ; Z ) + ( 2 / 2 ) F ( i , i ; 2 ; . = 0.
(5.134)
The expression inside the square brackets is easily seen to be zero by computing
the coefficent of z n . Thus we prove (5.124)(b).
We now verify (5.124) (c). The integrand in (5.132) is unaltered by multi
plication by x, and in place of (5.133) we obtain (1 — Z )~1 /2 — Z ~1
/3+[ K /4Z (1 —
z)i] F ( - | , | ; 2 ;1 - z). The analogue of (5.134) is
pi — I x m 2(x) dx Jo
; 2 ; * )
+ ( 7 T / 4 ) ( 1 - Z 2 ) / Z k i n -- [ ( l - z ) * / 3 . ] F |
'3 3
( 2 ' 2 : 1
^i n ^ l ^ i - z
1 3
2 ; 2 ;
(5.135)
To verify (5.124) (c) we then have to prove the following identity (by (5.118)
ril ,1:1:1-*
3 3
2 ' 2 '
+ < x / 4 ) [ ( l - * )2
/ * ]
1 3x F F - - , - ; 2 ; l - z .
2 ' 2 '(5.136)
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134 Group Invariance in Statistical Inference
From (5.131) with c = 1, a = b = §, then (5.129) wi th a = ±, 6 = - J , c = 1,
and then (5.130) wi th A = /i = 0, f = 1 we can rewrite (5.136) as
(4/3ir){l + 2»)
+ ( 3 z / 2 ) F ( i , - | ; 2 i * ) | , | ; 2 ; 1 - * ) + (4/3TT)(1 - z)
i i2' 2'
1 3
2 ' 2
(5.137)
Using (5.129) with a = - f , b = §, c = 1 + «r, and then (5.130) with n = 0,
c — « -* 0, the expression inside the square brackets in the last term of (5.137)
can be simplified to \ z F ( — I , ^ ; 2 ; r ) . Thus we are faced with the problem of
establishing the following identity
1 3
4/ TT = Fl — - , - ; 2; z j F\ - , - ; 1; 1 — z2 2
1 1
2 ' 2 '
(5.138)
which finally by (5.126) reduces to
-^FG'-^;2;1-2)+((1-£)/2-1)F(^;2;1-3) •
(5.139)
The expression inside the square brackets in (5.139) has a power series in 1 - z,
the value of which is seen to be zero by computing the coeificents of various
power of 1 - z. Hence we prove (5.124) (c).
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Some Minimal Test in Mul (in ormales 135
5.3.3. E-Minimax test (Linnik, 1966)
Consider the setting of Sec. 5.3.1. for Hotelling's T 2 lest for testing H 0 :
6 = 0 against alternatives Hj : 8 2 > 0 at level Q(T(0,1). The Hotelling T
2 test
0jv is £-minimax if
sup inf E8(<b)- inf Ee(<p N )<£ (5.139)
for all JV > /V 0 (e) where 8 = (ft, E ) and 0 runs through all tests of level < a.
Fo r N - * oo, A = ^ , 0 < tp < 2(log N ) * , (5.26) has the approximate
solution
which is a pdf on the simplex I V
If we substitute (5.140) in the left side of (5.62) we obtain a discrepancy
with its right side which is of the order of 0 £ ( J V - 1 + £ ) , for e > 0. Thus, if
a = ON satisfies the condition
0(exp{-logN)V*) < a < 1 - Ofex pf- l ogfV ) 1 ^) (5.141)
and we have
e x p ( - ( l o g 7 V } ^ £ ) < 8 < (logfiV)) 1 /* (5.142)
then
sup inf Ee(<t>) - inf Eg(4> N ) = O ^ l / i V 1 " 5 )
for e > 0 and hence the Hotelling T 2 test is e-minimax for testing H a against
i f , : 8 > 0.
E x e r c i s e s
1. Prove in details (5.65).
2. Prove (5.72) and (5.73).
3. Prove (5.119) and (5.120).
4. Prove (5.127) and (5.130).
5. Prove that in Problem 8 of Chapter 4 no invariant test under ( G T - , ! ^ ) is
minimax for testing Ho against Hi for every choice of A.
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136 Group Invariance in Statistical Inference
R e f e r e n c e s
1. M. Behara, and N. Giri, Locally and asymptotically minimax test of some multi
variate decision problems, Archiv der Mathmatik, 4, 436-441, (1971).
2. A. Erdelyi, Higher Transcedenial Functions, McGraw-Hill, N.Y ., 1953.
3. J. K. Ghosh, fnvariance in Testing and Estimation, Publication no SM 67/2,Indian Statistical Institute, Calculta India, (1967).
4. N. Giri, Multivariate Statistical Inference, Academic Press, N .Y . , (1977).
5. N. Giri, Locally and asymptotically minimax tests of a multivariate problem.,
Ann. Math. Statist. 39, 171-178 (1968).
6. N. Giri, and J . Kiefer, Local and asymptotic minimax properties of multivariate
test, Ann. Math. Statist. 39, 21-35 (1964).
7. N. Giri, and J. Kiefer, Minimal character of R 2-test in the simplest case, Ann,
Math. Statist, 34, 1475-1490 (1964).
8. N. Giri, J . Kiefer, and C. Stein, Minimax properties of T 2
-test in the simplest case, Ann. Math. Statist. 34, 1524-1535 (1963).
9. E . L. Ince, Ordinary Differential Equations, Longmans, Green and Co., London
(1926).
10. W, James, and C . Stein, Estimation with quadratic loss., Proc. Fourth Berkeley
Symp. Math. Statist. Prob. 1, 361-379 (1960).
11. E. L . Lehmann, Testing Statistical Hypotheses, Wiley, N .Y. (1959).
12. Ju. V. Linnik, V. A. Pliss and Salaevski, On Hotelling's Test, Doklady, 168,
719-722 (1966).
13. J . Kiefer, Invariance, minimax sequential estimation, and continuous time pro
cesses, Ann. Math. Statist. 28, 573-601, (1957).
14. Ju . V. Linnik, Appoximately minimax detection of a vector signal on a Gaussian
background, Soviet Math. Doklady, 7, 966-968, (1966).
15. M. P. Peisakoff, Transformation Parameters, Ph. D. thesis, Princeton University
(1950).
16. E . J . G . Pittman, Tests of hypotheses concerning location and scale parameters,
Biometrika 31, 20G-215 (1939).
17. O. V. Salaevski, Minimal character of Hotelling's T 2 test, Soviet Math. Doklady
3, 733-735 (1968).
18. J . B. Semika, An optimum poroperty of two statistical test, Biometrika 33, 70-80
(1941).
19. L . J . Slater, Confluent Hypergeometric Functions, Cambridge University Press(1960).
20. C. Stein, The admissibility of Hotelling's T 2 test, Ann. Math. Statist, 27, 616-623
(1956).
21. A. Wald, Contributions to the theory of statistical estimation and testing hypothe
ses, Ann. Math. Statist 10, 299-320, (1939).
22. A. Wald, Statistical Decision Functions, Wiley, N.Y. (1950).
23. O. Wesler, fnvariance theory and a modified principle, Ann. Math. Statist. 30,
1-20 (1959).
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138 Group Invariance in Statistical Inference
where q is a function on [0,oo) satisfying
J o(y'y)dy = 1
for y e E p .
We shall denote a family of elliptically symmetric distributions by E p(n, £).
It may be verified that
E{X) = u, cov(X) = o S
where a — p - 1 E{{X - n)'%-l (X - f*)). In other words al l distr ibutions in
the class E p{u t S) have the same mean and the same correlation matrix. The
family E p(ti, S ) contains a class of probability densities whose contours of equal
density have the same elliptical shape as the multivariate normal but it contains
also long-tailed and short-tailed distributions relative to multivariate normal.
This family of distributions satisfies most properties of the multivariate normal.
We refer to Giri (1996) for these results.
De f in i t ion 6.1.2. (Multivariate Elliptically Symme tric Distr ibuti on). A
n x p random matrix
x = (xij ) = (xl ,---,x n y
where Xi — (Xn,..., X ip )' is said to have a multivariate elliptically symmetric
distribution with the same location parameter p = ( p i , . . . , p p } ' and the same
scale matr ix S (positive definite) if its probability density function is given by
f x(x) = \L\-^ 2
q r g f o - tf^fa-tM , (6.2)
with Xi e E p ,i = 1,... ,n and q is a function on [0, oo) of the sum of quadratic
forms {ii — p ) ' S - 1 ( i i —p) , i — 1,.... ,H.
We have modified the definition 6.1.2. for statistical applications. Ell ip
tically symmetric distributions are becoming increasingly popular because of
their frequent use in filtering and stochastic control, random signal input, stock
market data analysis and robustness studies of multivariate normal statistical
procedures. The family E p(u, S ) includes, among others, the muiltivar iate nor
mal, the multivariate Cauchy, the Pearsonian type I I and type I V , the Laplance
distributions. T h e family of spherically symmetr ic distributions which includes
the multivariate-t distribut ion, the contaminated normal and the compound
normal distributions is a subfamily of i £ p ( p , £ ) .
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Locally Minimax Tests in Symmetrical Distributions 139
Write f x(x) in (6.2) as
(6.3)
where 8 = {p, E ) e Q,x e x and * is a measu rable function from \ o n t o 3̂ -
Let us assume that y is a nonempty open subset of R m
and is independent of
0 and q is fixed integrable function from y to [0, oo) independently of 8.
Let G be a group of transformations which leaves the problem of testing
H 0 :8 £ i l / / 0 against Bi : 8 £ flff, invariant.
Assume that \ is a Cartan G-space and the function i/j satisfies
for ai l i € x> ff £ G , # G SI an d g € G where (5 is the induced group of transformations on * corresponding to g £ G on x a r | d G acts transitively on
the range space y of *£.
Using (2.21a) the ratio i i of the prob ability density function of the maxima l
invariant T(X) under G on x; with 8\ 6 QH , , #o £ QH 0 > >s given by
where 6 (g) is the Jacobian of the inverse transformation x —> gx.
T h e o r e m 6.1.1. / / G acts transitively on the range space y of then R
is independent of q for all 8\ 6 tiff,, do £ fln0 -
P r o o f . Since G acts transitively on y , for any jo £ J there exists an
element k(x,8j : y 0 ) of G such that
Using the invariance property of p and replacing g by gh(x,8i : j / 0 ) we get
*(gx\8) = gV(x\8) (6.4)
(6.6)
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140 Group Invariance in Statistical Inference
where A r (-) is the right modular function of ft. Hence
Q{80 )8{h{x,80 : y0 )-i)A r(h(x,6>0 : y0 ))
is independent of g. •
6.2. L o c a l l y M i n i m a x T e s t s i n E p(fi, E )
In recent years considerable attentions are being focussed on the study of
robustness of commonly used test procedures concerning the parameters of
multivariate normal populations in the family of symmetric distributions. For
an up-to-date reference we refer to K a r i y a and Sinha (1988). T h e criterion
mostly used in such studies is the locally best invariant ( L B I ) property of the
test procedure. Giri (1988) gave the following formulation of the locally min
imax test in E p(ii, £) . Le t F be the space of values of the maximal invariant
Z — T(X) and let the distribution of T(X) depend on the parameter (<5,77),
with 6 > 0 and 77 g W. For each (u, 77) in the parametric space of the distribu
tion of Z suppose that f(z : $,tf) is the probability density function of Z with
respect to some r/-finite measure. Suppose the problem of testing Hg : 8 e f l / / 0
against Hi .8 e fi//, reduces to that of testing H 0 : 8 - 0 against Hi := A > 0,
in the presence of the nuisance parameter IJ , in terms of Z. We are concernedwith the local theory in the sense that f(z : A, n) is close to f{z : 0,17) when A
is small for all q in (6.2). Thro ugho ut this chapter notations like o ( l ) , o(h(X))
are to be interpreted a A —* 0 for an q in (6.2).
For each a, 0 < a < 1 we now consider a critical region of the form
R' - [U{x) = U(T(x)) > C a ] (6.7)
where U is bounded, positive and has a continous distribution for (8, r/), equi-
continuous in (S,ri) with 6 < S0 for any q (6.2) and R' satisfies
P 0 , „ ( f l * ) = cv, P M ( f T ) = a + h{\) + r fA, / , ) (6.8)
for any q in (6.2) where r(A,ri) - o(/ i(A)) uniformly in 77 with h{\) > 0 and
h(X) = o(l). Without any loss of generality we shall assume h(X) — bX with
£>> 0.
R e m a r k s .
(1) The assumption P0t „(R*) = a for any q implies that the distribution
of U under H 0 is independent of the choice of q. T h e test with critical
region R' satisfying the assumption is called null robust.
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Locally Minimax Teats in Symmetrical Distributions 141
(2) The assumption P\^(R") — a + h(X) + r(X,ri) for any q implies that
the distribution of U under Hi is independent of the choice of q as
X —> 0. The test with critical region R" satisfying the assumption is
called locally non-null as X —• 0.
Let fo, £x be the probability measures on the sets {S = 0}, {6 — X},
respectively, satisfying
//
f(z : X,T,)Udv)
= 1 + h(X)[g{X) + r{X)u] + B{z, X) (6.9)
f(z:Q, n )( 0(dri)
for any q in (6.2) where 0 < Ci < r(X) < c 2 < oo for X sufficiently small,
g(X) — o(l) and B(z,X) = o(X) uniformly in z.
Note. If the set 8 = 0 is a single point, £ 0 assigns measure 1 to that point. In
this case we obtain
j = t ( 6 1 0 )
Since G act s trans itively on the range space y of the right-hand side of
(6.10) is independent of q.
T h e o r e m 6.2 .1 . If R* satisfies (6.8) and for sufficiently small X, there
exist £\ and £o satisfying (6.10), then R* is locally minimax for testing HQ :
5 = 0 against the alternative Hi : 6 = X for any q in (6.2) as X —* 0, i.e.
l i m iaU,Px,v(R*) - a = j , f i ^
A - . o s u p ^ i e Q a iaf„Px,v{4x rejects H 0 } - a
for any q in (6.2), where Q a is the class of test d>\ of level a.
P r o o f . Let
n = (2 + /i(A)[9(A) + C*Q7-(A)])- 1 .
A Bayes critical region for (0, 1) losses with respect to the a priori £ J —
T\ Z\ + (1 - TX)£O is given by
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142 Group Invariance in Statistical Inference
B y (6.10) B x(z) holds for any q in (6.2). Write B x(z) = B x , V x = R * - B x
and W x = B\ — R". Since supJ5 A (z )/ 7 i ( A ) | = o( l) and the distribution of U
is continuous, letting
KM) = / PxMUdv)
we get
Since, with A — Vj, or
p & t ^ ) = % f , 4 ! ( i + o ( f t ( A ) )
writing
r J ( A ) - (1 - r A ) P J i A ( A ) + r A ( l - f & f A ) )
we get the integrated Bayes risk with respect to fj as
rJ(B>) - (1 - r O P o , x ( S , ) + r A ( l - P & p j J ) .
r x(R*) =(1 - r A ) P 0 % ( J ? ) + rx(l - P ; , A ( P ' ) ) •
Hence for any q in (6.2)
r A ( B A ) = rl(R-) + (1 - n ) t * f o T O "
+ T a ( P 1 ' I A ( V A ) - P 1 - I , ( W , ) )
= rl(R') + (1 - 2 n ) ( P 0 * A ( H M - P 0 * A ( V A ) )
= r i ( i l * ) + 0 ( M A ) ) . (6.13)
If (6.11) is false for all q in (6.2), then by (6.8) we can find a family of tests
{<px} of level a such that d> x has power function cv + r(A, n) on the set {6 = A}
for all q in (6.2) satisfying
lim sup (inf[r(A,jj) -h(X)]/h(X) > 0.
A —> 0 "
Hence, the integrated Bayes risk rA
of d>\ with respect to then satisfies
lim sup ( r ; ( P ' ) - r A ) / M A ) > 0
A — 0
for all q in (6.2) contradicting (6.13) •
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Locally Minimax Tests in Symmetrical Distributions 143
6 . 3 . E x a m p l e s
E x a m p l e s 6.3 .1 . Let X = (Xij) = (X[,. .. t X' n ), X[ = (X' tl ,... ,X ip ),
i — 1,... ,n be a n x p (TI > p) matrix with probability density function
f x(x) = m-^ 2
q(Y (xi ~ rf'E-^-JK)) (6.14)
i-i
where g is from [0, co) to [0, oo), ft = (pi,... ,p p )' € E p and E is a p x p positive
definite matrix. We shall assume that q is thrice contunuously differentiable.
Write for any b = (h,...,b p )' = {b'{1) ,b'(2) )', with b{ i) - (f>i,... , 6 P l ) ' , b [2 ) =
(lfa+1,.. .»6p)',6p] = (bi,...,bi)' and for any pxp matrix
A=(aiS )=(f*
3 \ A 2 1 A 22
where A{ 3 are pi x p } submatrices of A with pi -I- p 2 — p- We shall denote by
/ a n , . . . , a i , \
\aa,..., On/
We are interested here in the following three problems.
P r o b l e m 1. To test H IQ : p = 0 against H \ \ ; p ^ 0 when E is unknown.
P r o b l e m 2. To test H 2I) : M(i) = 0 against H 2 \ • P(i) ^ 0 when p( 2 ) , E
are unknown.
P r o b l e m 3. To test H Z Q : p — 0 against H 31 : p; ij = 0, p( 2 ) / 0 when E
is unknown. The normal analogues of these problems have been considered in
Chapter 4.
P r o b l e m 1.
Let Gi(p) be the mul tipl icative group of p >c p nonsingular matrices g.
Problem 1 remains invariant under G;(p) transforming
(X,S;p,-i:)^(gX,gSg';gp,gT l g')
where X = ^ = l X i , S = Z?=l (Xi-X)(Xi-X}'. A m axi ma l invariant in the
space of ( X , S) is T 2 = nX'S^X' or equivalently R = nX'(S + nXX')~1
X =
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144 Group Invariance in Statistical Inference
T 2 / ( l + T 2 ) . A corresponding maximal invariant in the parametric space under
Gi(p) is 6 = T t y t ' E ~ V - If Q is convex and nonincreasing, then the Hotelling T 2
test which rejects H\o whenever i i > C is uniformly most powerful invariant
with respect to G ; (p) for testing i f 1 0 against H\i (Kar iya (1981}}. As stated in
Chapter 5, the group Gi(p) does not satisfy the conditions of the Hunt-Stein
theorm. However this theorem does apply for the subgroup GT(P) o f p x p
nonsingular lower triangular matrices with p > 2. From Example 5.1.1. the
maximal invariant under GT{P) is [R\,... , i?p) ' , whose distribution depends
on A — (6\,...,Sf,)'. The ratio R of probability densities of (Ri,...,Rp)'
under Hu : S = A and H L 0 : 6 = 0 is given by (with g = (gij) € GT = GT{P))
j Pi(9z)fl(9 2
i)(
"-i)/2
d9
R = ~ — ^ (6.15)
/ P o ( s x ) I ] ( ^ ) ( n - 0 / 2 d S
where
dg = Jldgij,
Pi(x) = q(tr x'x - 2nxp + A),
Po(x) = ?(tr x'x).
Since x'x > 0, there exists a g E GT such that
gx'xg' = I, Jn~gx = y = (VR U ..., v ^ p ) ' .
Hence
R =
f q(ti{ 9 g' - 2A'gy + tyf[(fi)f**&*B
/ 9 ( t r ( s s ' ) ) n ( ^ ) ( n - i ) / 2 d f f
(6.16)
Under the assumption that q is thrice differentiable, we expand q as
q(ti(gg' - 2A'gy + A))
- 1(^(99')) + ( -2tr(A'gy) + \)q™(ti gg1 )
+ ^ 2 t r ( A ' g y ) + A) 2 ( tr (gg'))
+ h-2ti(A'g y ) + X) 3q 3(z) (6.17)
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Locally Minimal Tests in Symmetrical Distributions 145
where z = ti{gg') + (1 - a ) ( - 2 t r ( A ' 9 y ) + A) , 0 < c < 1 an d -
Since the measures
9%*<^)>jl^^^dfl , fc = 1.2,3i=i
are invariant under the change of sign of g to -g, we conclude that
/ t r ( A W " ( t r ( S 9 ' ) ) fiftfe**-*** = ° '
/ 9 i W t ) ( t r ( 9 9 ' ) ) n ( 4 ) ( ' 1 " i l / 2 r f ^ 0 (618)
for i j£ i , j m . Now letting
D = / g ( t r ( f f 0 ' ) ) n ( s ? i ) ( u - ° / 2 ^ ,
from (6.15)-(6.18) we get
R=l + ± ! g^(gg'))f[(gir-^dg
+ 4 / M A ' 0 i , ) ) 2
9 ( 2 > ( t r ( 5 f f ' ) ) n ( ^ ) | n - ' ) / 2 d S
+ 4 / ( - 2 t r ( A '? s /
) + A ) V
3 1
(
3
) n ( S
2
, )
| n
-
l ) / 3
^6 / ? i i (6 .19 )
The first integral in (6.19) is a finite constant Q\. To evaluate the second
integral in (6.19) we first note that
tr{L\'gy) = Y d r)12 (6.20)
From (6.17) and (6.19) the second integral in (6.18) reduces to
/ | 5 > £ M - I ^(gg'))f[(gl^-^dg. (6.21)
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Locally Minimax Teats in Symmetrical Distributions 147
where
/ P M = E[ T J ( E I ) ( « - 0 / a
\ i = l
C = p(p + l ) / 4 + 1/2 £ ( n - i ) . (6.26)
Let JJ = ( i j i , . . . ,%,) ' , r?i — ^i/A. From (6.18) we get
» = X-+ A2? j&+ | J |j> + ( n - j +
L
)%]j j +(6.27)
where B(y,ri,\) — o(A) uniformly in and 77. Thus, from (6.27), with
the equation (6.9) is satisfied by letting £0 give measure one to the single point
?7 = 0 while £A
gives measure one to the single point 77—
n" — ( j j j , . . • , t j j )whose j t h corrdinate Wj satisfies
V'j = (" " j ) " V " 3 + i r V ^ f f t " P). / = 1, - (6-28)
so that
+ l ) n | - njp. (6.29)
Since G;(p) and GT (p) satisfy the condition of Theorem 6.1.1., R does not
depend on q. The first equation in (6.8) follows from the following lemma.
L e m m a 6.3.1. Let Y = (Y u ... ,Y n )', Y; = {Yn,..., Y ip )' f i = 1 , . . . , « be
annxp random matrix with spherically symmetric probability density function
f Y (y) = 9(tr yy') = q ( g tr ] , (6.30)
a n d Jet
, i = l
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148 Group Invariance in Statistical Inference
where Y ( L ) , Y( 2 ) are of dimensions k x p, (n - k) x v respectively and n - k >
p>k. The distribution of Y ( 1 | ( Y ( ' 2 ) Y ( 2 ) ) Y (1) does not depend on q.
Proof. Since n > p, Y'Y is positive definite with probabili ty one. Hence
there exists a p x p symmetric positive definite matrix A such that Y'Y — AA.
Transform Y to U, given by Y = UA. As the Jacobian of the transformation
Y — U is \A\ n
,
/ r / ,A(« ,«) = «(tr a a ' ) M n .
Thus (7 has the uniform dis tribution and 17 is independent of A. Partition
U as
where t/ ( 1 ) is k x p and (7(2) is (n - k) x p. From Y = C M we get Y ( { ) = t7#j>4,
i = 1, 2. Thus
Since Y ( 1 1 ( Y 1 ' 2 ) Y ( 2 ) ) _ 1
Y { j j is a function of 17 alone, it has a completely specified
distribution. •
C o r o l l a r y 6. 3.1. If fc- is distributed as Hotelling's
T 2 (central i.e. 6 = 0).
Proof. Since Y( 1 j (Y j ' 2 j Y (2 ] )- 1
Yj ' 1 j has a completely specified distribution
for all q in (6.30), we can, without any loss of generality, assume that
Y i , . . . , Y„ are independently distributed N p(0,I). This implies that Y / 2 j l ( 3 ) =
E J L 2 Y / Y / has Wishart distribution W p(n - 1,1) independently of Y i . Hence
Y
(i)( Y
(2)Y
m)~lY
{i)h a s
Hotelling's T 2
distribution. •
From this it follows that when 6 = 0, T 2
= nY'S'1?, with Y = l/n £"=1
Yi , 5 = ~Y
)( Y
i ~Y
Y h a s Hotelling's T
2 distribution for all q in
(6.30).
To establish the second equation in (6.8) we proceed as follows. Us ing
(2.21) we get for any g £ t?/(p), (which we will write as Gi)
fT*{t 2 \6) J Gi g(tT( 9 g'-2a'gy + S))\gg'\^' 2 dg
fT2(t 2
\0) L <j(tr gg')\g0«-»)/ 2 dg( J
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Locally Minimax Testa in Symmetrical Distributions 149
where y = ( < /r ,0 \ . . . , 0 ) ' , a - ( \ A , 0 , . . . ,0) ' are p-vectors and T = f 2 / ( l + f 2 ) .
After a Taylor expansion of q, the numerator of (6.31) can be written as
/ q(ti99')\99'\iin- p, dg + 6 j <,<»>(tr 99' ) \99 ' \ { n - p ) n dg JG, JG,
+ 26r f q^(tT gg , )gl\gg'\^->'y 2 dg + o(6) JG,
where g — (jJy). Using (6.18) we get the ratio in (6.13) as
l + 6(k + CT) + B(y,t),6) (6.32)
where B(y,r),6) — o(6) uniformly in y,v and 77 — ( i j i , . . . , i7 p ) ', with r/i —
Si/S,i = 1,.. . ,p and k,c are positive constants. Hence for testing H l0 : 6 = 0
against the alternative H' n : S — A the Hotelling's T2
test with critical region.
R* = {U(X) = R>c a } (6.33)
where c a is a positive constant so that R* has the size a, satisfies
P^tRn^a + bX + giKr,) (6.34)
with b > 0 and <7(A,r/) = o(Z>A). Thus we get the following theorem.
T h e o r e m 6.3.1 . For testing Hm : n = 0 against the alternative H' n : 6 —
A specified > 0 Hotelling's T 2 test is locally minimax as X —* 0 for the family
of distribution (6.14).
R e m a r k 1, Since Gj(p) satisfies the condition of Theorem 6.1.1., the ratio
(6.31) is independent of q. From Corollary 6.3.1. and equations (6.31) and
(6.34) it follows that Hotelling's T 2 test is locally best invariant for testing
Hia against : 6 — X as A —* 0 for the family of distributions (6.14). I f qis a nonincreasing convex function then Hotelling's T 2 test is uniformly most
powerful invariant.
P r o b l e m 2. Write X = ( X ( i j , X ( 2 ) ) with : n x p\,X( 2 ) : n x P2, and
p,+P2= p.
Since the problem is invariant under translations of the last pa components
of each Xj, j — 1,... ,n, this can be reduced to that of testing if 2
o : p ^ j — 0against H21 • P( i) 0 in terms of X^ only whose marg inal probability density
function is given by
/ ^ ^ ( ^ D i - I S i i r ^ g f t r S r / ^ i j - e p ' ^ y ^ ^ - e p ^ ) )
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150 Group Invariance in Statistical Inference
where E i j is pi x P l left-hand cornered sub mat rix of £ , e — (1 , . . . ,1)
JI-vector and
g(tr v) = / q(tr(v + ww'))dw,
J R "F1
and q is a function on [0, oo) to [0, oo) satisfying (6.1a). Now, us ing the resul ts
of problem 1 with p = p L , R = Ri - R i w e h a v e t h e following theorem.
T h e o r m 6 .3 .2 . For testing H20 • P(i) = 0 against H' 21 : 6\ — np'^y
S n V ( i ) = X {specified) > 0 the test which rejects H 20 for large values of
Rj — nX'(1)(S{n) +r$( l y£fa)~1 Jt(i)
= nXl l)S( l \ ) X {l) /{l + nXl 1)S-\ ) X l , ) )
is locally minimax as X —t 0 for the family of distributions (6.34).
R e m a r k 2. T h e locally best invariant property and the uniform ly most
powerful invariant property under the assumption of nonincreasing convex
property of q follows from the results of problem 1.
P r o b l e m 3. T h e invariance property of this problem has been discussed
in Ch ap te r 4. T h e problem of testing i f 3 0
against H 3
i remains invariant underthe multiplicative group of transformations G of p x p nonsingular matrices g,
9 = ( 9 m ) ° \9{2i) 9(32)
where gin) is pi X P i , transforming (X, S) -» (gX, gSg'). A ma xim al invariant
in the sample space under G is (Hi,.S3) where
pi p
Ri=2^Ri, Ri+R~ 2 = R = Y, Ri
-fc=l i = l
A corresponding maxim al invariant in the para metri c space is S\, 62 where
i = l
P
i - l
For invariant tests under G the problem is reduced to testing i i 3 0 ; 5 2 = 0
against the alternatives H 3l : b\ > 0, when it is given that 61 = 0, in terms of
Ri and R 2-
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Locally Minimax Tests in Symmetrical Distributions 151
As stated in Chapter 5 the group G does not satisfy the conditions of
the Hu nt -S te in theorem. However this theorem does apply for the subgroup
GT ( P) of p x p nonsingular lower triangular matrices. From problem 1 the
maximal invariant under GT{P) is (Ri,...,Rp)' whose distribution depends
continuously on A = , • - •, VS P
)' with 6i > 0 for all i. UnderH
30
: 6i — 0,i = l , . . .p and under i/31 : 6i — 0, i — 1 , . . . , pi- The ratio R of the probability
densities of (Ri,..., R p )' under H ' 31 : 6 2 = A > 0 and under H 3 Q '• $2 = 0, when
it is given that 6\ — 0, is given by
f
/ q(tv(g 9' - 2A{ 2j(g{2im) + S ( 3 2 ) J , ( 2 ) + A)} ]\[gl){n- ,)l2 dg
R = — — 1 (6-35)
/ ?(tr gg')\{{gir-^dg
J G T 1
where g e G T (p) — GT , y — ( T/R~I , • • • ,VRy)' and
• - * • > - ( a a )with (» i i j ff (22) both lower triangular matrices .
As before we expand 9 in the numerator of (6.35) as
9(tr Sff') + (~2v + \)qM(ti gg') + {-2v + A ) 2
9 | 2 ) ( t r 3 0 ' ) + ( -2 f + A ) V 3 , ( ^ ) (6-36)
where
2 = tr gg' + (1 - a ) ( - 2 f + A), 0 < a < 1,
" = tr A {2 )
( 9[ 2 1
) 3 /( 1
) + 3(22)2/(2))-
Using (6.18) the integration of the second term in (6.36) with respect to
the measure
n ^ - ^ ( 6 - 3 7 >1
gives AQI where
0 1 = / qW (K99')f[(9 2 ,)in-iy2 dg.JG T 1
To integrate the third term in (6.36) with respect to the measure given in
(6.37) we first observe that (by (6.18))
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152 Group Invariance in Statistical Inference
and
Denote
/ ( t r ( A ; 2 ) S ( 2 u y o ) ) % ' 2 H t r f f s 0 n ( 5 2 J l n - , ) / 2 ^
= / ( E 9 ( 2 t ( t r S f f ' ) n ^ ) ( n " i ) / a d f
/ ( t t ( A ; 2 ) f f ( 2 2 ) i ( ( 2 ) ) % ( 2 > ( t r S 3 ' ) n ( s 2
1 } ( n - 0 / 2 r f sJ G r
v i
= / E r >Ei f f 4 + - + V i I )T
[ )=Pi + l j > i
1
K= f q^[tigg')dg,JG T
£ = tr gg',
D= f 9(tr gg')f[{glY-^dg,
where e-s are defined in (6.23). From (6.35)-(6.40) we get
D
+ B{y,r,, A)
( 2MN ^ ( n+ E r i
1>J
(6.38)
(6.39)
(6.40)
(6.41)
where B(y,n, A) — o(A) uniformly in j / , T> (independently of q by Theorem
6.1.1). The set {A = 0} is a simple point 17 = 0. So the £ 0 in Theorem 6.2.1.
assigns measure 1 to the single point rj = 0. The set {<52 — A} is a convex
P2 = (p — pi) dimensional Eucl idean set wherein each component 77; = 0(h(X}).
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Locally Minimax Tests in Symmetrical Distributions 153
Any probability measure £ \ can be replaced by degenerate measure which
assigns measure 1 to the mean 77*, i — pi + 1 , . . . , p of £ j . Hence
p
+ B(y,ri,\) f g 4 2 j
where B(y,Tj, A) = o(h{\)) uniformly in y,n. Consider the rejection region
RK = {X : U(X ) = fi + Kf 2 > c0 ) (6.43)
where A" is a constant such that (6.42) is reduced to yield (6.9) and C„ depends
on the level of significance a of the test for the chosen K, independently of q
(by Lemma 6.3.1.) Now choose
„ _ (n-j-!)••• (n-p) I n-pi \ , _
V j (n-j + l) --(n-p + 2) {(n-p+Vpi) ' 3 P l
so that
Hence we can conclude that the test with rejection region
R' = ^x :U(x)=f 1 + 7 ^ ^ - r 2 > C 0 J (6.45)
with P(,TTI (R') = a satisfies (6.9) as A — > 0. Furthermore any region RK of the
form (6.43) must have K = to satisfy (6.9) as A - * 0 for some t\.
It can be shown that (Giri (1988))
ffi, R 2 (?i, f 21 J?M )//fi, (fi, f 2 \H 3a )
Dl\P2 \ \
P2 (6.46)
where Dl7 ati, a 2 are positive constants and B{f '1,^2,62) — 0(62) uniformly in
f j , f 2 . From this and Theo rm 6.1.1 it follows tha t for testing H 3D against Jf 3 1
the test with critical region R' is locally best invariant as 6 2 —> 0. Hotelling's
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154 Group Invariance in Statistical Inference
test which rejects H 3Q whenever f i + f 2 > constant does not coincide with the
L B I test and hence it is locally worse. From (6.32) it follows that Hotelling's
test whose power depends only on 6 = 6 2 , has positive derivative at 6 2 = 0.
Thus, for the critical region of Hotelling's test the value of h(\) with 6% = A in
(6.8) is positive. The first equation in (6.8) follows from Lemma 6.3.1. Hence
we get the following theorem.
T h e o r e m 6.2.2. For testing H 30 against H 3l : £ ( 2 ) = A the test, which
rejects H 30 whenever fi + ^jf-rt > constant, is locally minimax as A —* 0 for
the family of distributions (6.2).
E x a m p l e 6.3.2. Let X = (X I} ) = {X\ X n )' X[ = (X* X ip )',i =
1,..., TI be an n x p random matrix with probability density function given in
(6.2). Let X = J Xt, S = J J ( X i - X)(X t - X)'. For any p x p matrix A
we shall write
C-4(l l | -4(12) 4 ( i 3 ) \
Ajai) A(2 2) -4(23) I (6-47)
4(31) 4(32) .4(33) /
with 4 ( n ) : 1 x 1, -4{2
2) I Pi xp i , 4 (3 3
, : p2 x p; so that pi +pa = p—
1. Define
Pi =S ( 1 2 ) S ( 2 2 ) S ( 2 1 ) / S ( I 1 ) '
p2
= p2
+ p 2 - ( E , i 2 ) , E , i 3 , ) ' ( ^ ( E a a , , E ( 1 3 ) ) ' / E ( 1 1 ) .
We shall consider the following three testing problems: (see Sec. 4.4)
P r o b l e m 4. To test H 0 : p 2
= 0 against Hi : p 2 > 0 when p, E are
unkown.
P r o b l e m 5. To test H 0 : (? = 0 against H 2 : pi > 0, p\ = 0 when p , E
are unknown.
P r o b l e m 6. To test H 0 : p2
= 0 against H 3 : p\ — 0, p\ > 0 when p, E
are unknown.
These problems remain invariant under the group of translations trans
forming
( * , S ; i » , S ) - » ( * + 6 , S ; / £ + t , E ) . 6 e Rp
.
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Locally Minimax Testa in Symmetrical Distributions 155
The effiect of these transformations is to reduce these problems to correspond
ing ones where p = Q and 5 = ^27 =1 XiX[. Thus we reduce n by 1. We treat
now the latter formulation with p = 0 and assume that n > p > 2 to assert
that S is positive definite with probabil ity one. Let G be the full linear group
of p x p nonsingular matrices g of the form
f s m 0 0 \ 9 = \ 0 9(22) 0 . (6.48)
\ 0 9(32] 5 ( 3 3 ] /
For the first problem p 2 = 0, p\ =p — 1 and for the other two problems
Pi> 0,p2 > 0 satisfying p\+p2 — p— 1- T h e problems above remain invarian t
under G operating as
( S ; E ) -> (gSg'^Zg1
)
with g G G. Fo r Pro ble m 6 a ma xima l invariant in the space of S under G
(with p2 = 0) is
_o 5
( 1 2 )5
( 2 2 )5
[ 2 1 i
" =
Q
and a corresponding maximal invariant in the parametric space of E under the
induced group (with p2 = 0) is
p* = S ( 1 2 ) S
^S f 2 1 ) . (6.49)
s
( n )
For Problems 5, 6 a maximal invariant under G is (fi 2 , fi2), where
R\ = 5 " 2 , y 2 ^ ( 2 1 ) , (6.50)6
( H )
fij+fi| = ( 5 ( 1 2 ) , 5 ( 1 3 1 ) ( ^ l ( % a ) , •
(6.51)
A corresponding maximal invariant in the parametric space of E is {p\,p 2 ),
where
Pi =
., _ E ( i 2 ) E ( 2 2 ) E ( 2 i ]
E ( n )
( E ( 1 2 ) , E ( 1 3 ) ) ( ' ^2 2
) ^3
) ) ( S ( 1 2 ) 1 E ( 1 3 ) ) '
L ( u )
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156 Group Invariance in Statistical Inference
P r o b l e m 4. T h e invariance of this problem has been discussed in Sec. 4.4.
The group G does not satisfy the conditions of the H unt -S te in theorem . T h e
subgroup GT of G , given by,
f ( a m0 0
GT = < 9 - 0 0(22) 0
( \ 0 9(32) 9(33)
where 3( 22)(Pi x Pi)> 9(33)(P2 x Pi) are nonsingular lower triangular matrices,
satisfies the conditions of the theorm. It may be rem arked that we are using
the same notation g for lower triangular matrices as g in G . In the place of fi2,
the m ax imal invariant in the sample space under GT is a (p — 1)-dimensional
vector R = ( R 2
, R p
) ' denned in Exa mpl e 5.1.5. wi th
PI + I P
J=2 j = 2
The corresponding maximal invariant in the parametric space under GT is
A — {6^ 2 , • • • ,t>)J 2
)' as defined in Ex amp le 5.1.5. with
P I + I p
p\ = P 2 =P 2i+pl = J26i
i=2 i - 2
For testing H 0 : p 2 = 0 against the specified alternative Hi\ : p 2 = A the rat io
fi of the probability density function of R under H \ \ to that under HQ (using
Theorem 2.7.1.) is
R = fR(r\H lx
)/f R
(r\H 0
)(1 - A ) " " / 2
/ $((1 - A) hT(g 2
u) - 2g(11) Ay'g'(22) + g{ 22) g( 2 2)]v{dg)
(6.53)
where
•=2
= / g(tr gg')v(dg). JGT
i=2
lGT
y = ( v ^ 2 y/Tp)'
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Locally Minimax Testa in Symmetrical Distribntxona 157
Since the (p - 1) x (p - 1) matrix
C = ( l - p 2 ) " 1 ( / - A A ' )
is positive definite there exists a (p— 1} x (p— 1) lower triangular nonsingularmatrix T such that TCT' = I. Define 7i,a<,2 < i < p by
i
7i = 1 - E '̂' ^ = &7j>/7<7i-i-
j = 2
Obviously 7 P = 1 - p 2 . Le t a = ( a 2 ) . . . , a f ) ' . The n a = T A . Since C A = A
and TCT' = I we get
a = T A - r C A = { r ) _ 1 A ,
a'ot = A ' ( T ' T )- 1
A = A C A '
= (1 - p 2 ) - 1 ( A ' A ) - ( A ' A ) { A ' A ) = p 2 .
Furthermore with = (a 2 , • • •, on)'
i
Since
| C | - ( l - p 2 } 2 - "
we can write
(1 — A ) " ' " / 2 ' f
M=- g g(tT(9 2
U )^ 2 9(ii)ay'9(22)+m2)9{22)))l/ ( d 9)- (6-54)
Expanding the integrand in (6.45) as
9(tt 90') + o ( 1 ) ( tr S<7 ' ) ( - 2t r ( a ( 1 1 ) C ! i , ' o ; 2 2 ) ) )
+ ^ ^ ( - 2 t r ( f f l l l ) Q ^ 2 2 ) ) ) 2
+ ^ ^ { - 2 t r ( g , n | a y ' g ( 2 2 | ) )3
where
z = tr go' + (1 - ff)(-2tr 9 ( i i | a y ' S ( 2 2 ) ) '
with 0 < S < 1.
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158 Group Invariance in Statistical Inference
Since, with g - (gtj),
P
trg (ii,QJ/'<i(22] = tf(ii> E f o ) I / 2 ( + a
*9 i
j ) j=2 i>j
and the measure g (1> (tr gg') f(do) is invariant under the change of sign of g
to — g we conclude as in (6.18) that
/ [tr <J(i i )Q!/ ' i7(22)]9 ( 1 ) ( t r 99'Hdg) = 0 JGT
j 9i j 9ihQ(1)(^gg')u(dg) = 0! i / /, J j t f e . (6.55)
Let
L = tr gg', L e i = gj u) , L e , = g2
, i = 2 , . . . ,p,
L e 2 p - i + i = G2
+ 2 , i , i = 2, . . . , p ,
i e i + P ( p - i ) / 2 = ffp, •
Now proceeding as in Problem 3 we get
j = 2 • \ !>j
+ £ ( > / , A, r>)
where
I f =6j/X, j = 2 , . . . , p ,
M = E
j=2
e ( n - i ) / a T T J " - * *1
) / 2
(6.56)
S ( j / i , A , n ) = o(A) uniformly in 3/ and TJ. Furthermore from Theorem 6.1.1 R
does not depend on q.
Since A = 0 is a single point n = 0, £0 assigns measure 1 to the point 17 =
0. The set p2
= A is a convex (p —1)-dimensional Euclidean set wherein each
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Locally Minimal Tests in Symmetrical Distributions 159
component is 0(A) . Thus any probability measure & can be replaced by the
degenerate measure which assigns measure 1 to the mean TJ- , i = 2 , . . . , p of t)y •
Choosing
V* = (n-j + l ) - ' ( n - j + 2 ) ~ 1 ( p - l)~l n(n - P + 1 ) ,
f = 2 , . . . »J> . (6-57)
we see that (6.9) is satisfied with U = YZ =2 R 3 = R 2 - F l o m G i r i ( 1 9 8 8 ) o r
using Theorem 2.7.1. we get
/ ^ ( r 2 | 7 J i ) / / f i 2 ( r 2 | f f 0 )
(1 - p 2 )-' 2
B„2i
y G
+ ( i - p 2 ) ( / - A A ' ) 9 ( 2 2 | 0 ; 2 2 ) ] ) p ( r f 9 )
where g e G , as defined in (6.48) , wit h p 2 = 0,
^ S ) - ( 9 ( 2 n , ) ( " - , l / 2 | f f ( 2 2 ) 9 f 2 2 ) |n ( p - 1 , / 2 ^ ,
5 = / g(tr gg')p{dg),
and u - (tii> - - • > T p - i ) ' satisfying u'u — r 2 . I f p 2 is small the power of the
test against the alternative H\ which rejects Ho whenever R 2 > C a is given
by a + h{p 2 ) + o(p 2 ), where h{p 2 ) > 0 for p 2 > 0 and h(p 2 } = o( l ) . The
null robustness of the test follows from the fact that T(X) = R 2 satisfies
T(X) = T(Xh) where h 6 GT (Kariya (1981b)). Hence we get the following
theorem:
T h e o r e m 6.2 .3 . For testing H a against H[: p 2 = A, the level-a test which
rejects Hn whenever R 2 > C is locally minimax as A —> 0.
Problem 5 . H ere
m = 0, j = pi + 2 , . . . , p .
For testing H Q against H 2 - p\ = A we can write (using 6.55)
- TlA R = l + Y
•pi+i
3=2 M>j / .
+ B(r,X,n). (6.58)
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160 Group Invariance in Statistical Inference
Now letting give measure one to n = 0 and t\x give measure one to
Vh---< n p,+i'tlie m e a n ot*£> where
j ) | = ( n - p 1 ) 7 I ( 7 i - i + l ) - 1 ( n - J - l - 2 ) - 1 p r 1
J = l , . . . , P i . (6.59)
we conclude that (6.9) is satisfied with U = Rf = £,,'=2R j a n d ( 6 - 8 ) follows
from Theorem 6.2.3.
T h e o r e m 6.2.4. For testing Ha against H' -p\ — A, the level-a test which
rejects Ho whenever Rf > C a , where the constant C 0 depends on level a of
the test, is locally minimax as X —* 0.
Problem 6 . Here
T ) , = 0 , j = 2 , . . . , p i + l .
For testing Ho against H 3 ; p\ = X we can write R as A —< 0 as
)=P i+2 V i »2 B
+ B ( r , A , n ) (6.60)
where B(r, X, n) = o(A) uniformly in r and rj. Let us now consider the rejection
region of the form
RK ={? ; u(y) = f\+Kf 2 2 > c a ] (6.61)
where K is chosen such that (6.61) is reduced to yield (6.9) and C„ depends
on the level a of the test for the chosen K. Let Co give measure one to n = 0
and t]x give measure one to a single point ( 0 , . . . , 0, r v * i + 2 , . . . , JJp) where
Tl'j=(n-p+ l)n(n - j + l ) " l ( n - j + 2 ) - \ ( p - 1 ) " \ }=Pi+2,...,p
and let
R=! [ y:U(y) = fl + ^ f l > C a }
with pi + P2 = p - 1. Observe that the rejection region R with P0> „R = a
satisfies (6.9) . Fu rthe rm ore, for the invar iant region R, PxiTI {R) depends only
on A. Hence, from (6.8) r ( A , r ) = 0. Since R is locally best invariant as A -*
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Locally Minimax Testa in Symmetrical Distributions 161
0. the test, which rejects HQ whenever R 2 > C, does not coincide with R and
hence is locally worse. From Problem 4 the power of the test (which rejects
H 0 whenever R
2 > C) which depends only on A, has positive derivative at A
= 0. Thus from (6.8) with R* = Rwe conclude that h(X) > 0. Hence we have
T h e o r e m 6 .2 ,5 . For testing HQ against H ' 3 : p\ = X > 0, the test with
critical region R is locally minimax as A —> 0.
E x e r c i s e s
1. Let X = (Xi,.... X p )' be a random vector with peobability density function
(6.1). Show that E(X) = p, c o v ( X ) = Q s with a = p~l E(X - / i J ' E "1 (X-
!*))•
2. Let X be a n x p random matrix wi th probability density function (6.2) .
Find the maximum likelihood estimator of p and E .
R e f e r e n c e s
1. N. Giri, Locally minimax tests for multiple correlations, Canad. J. Statist., pp. 53¬
60 (1979).
2. N. Giri, Locally minimax tests m symmetrical distributions, Ann. Inst. Stat. Math.,pp. 381-394 (1988).
3. N. Giri, Multivariate Statistical Analysis, Marcel Dekker, N.Y. (1996).
4. T. Kariya and M. Eaton, Robust test of spherical symmetry, Ann. Statist., pp. 208¬
215 (1977).
5. T . Kar iya, A robustness property of Hotelling's T 2-problem, Ann. Statist., pp. 206¬
215 (1981).
6. T. Kariya, Robustness of multivariate tests, Ann. Statist., pp. 1267-1275 (1981b),
7. T . Kari ya and B . Sinha, Non-null and optimality robustness of some tests, Ann,
Statist., pp. 1182-1197 (1985).8. T. Kariya and B. Sinha, Robustness of Statistical Tests, Academic Press, N.Y.
(1988).
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Chapter 7
TYPE D AND E REGIONS
The notion of a type D or E region is due to Isaacson (1951). Kiefer
(1958) showed that the usual F-test of the univarial general linear hypotheses
possesses this property. Lehmann (1959) showed that, in finding type D region,
invariance could be invoked in the manner of the Hunt-stein theorem; and this
could also be done for type E regions (if they exist) provided that one works
with a group which operates as the identity on the nuisance parameter set H
of the testing problem.
Suppose, for a parameter set fi = {( 8 ,-n) '• 8 € 0 , J J e H with associated
distributions, with 0 a Euclidean set, that every test function d> has a power
function fy(9,IJ) which, for each n, is twice continuously differentiable in the
components of 6 at 8 = 0, an interior point of 0. L e t Q a be the class of locally
strictly unbiased level a test of HQ : 8 = 0 against Hi : 0 ^ 0. Our assumption
on 0$ implies that all tests in Q„ are similar and tha t d0^/d&i\g ̂o — 0 for all d>
in Q a . Let A^(JJ) be the determinant of the matrix B<j,(rj) of second derivalives
of 0${8,TI) with respect to the components of 8 (the Gaussian curvature) at
8 — 0. Suppose the param etrization be such that A # ( f i ) > 0 for all n for at
least one <(>' in Q a .
D e f i n i t i o n 7 .1 . T y p e . E t e s t . A test d>' is said to be of type E if <p' e Q a
and A ^ . ( n ) = max^gp,, A ^ ( n ) for all 77.
D e f i n i t i o n 7 .2 . T y p e D test . A type E test d>' is said to be of type D if the nuisance parameter set H is a single point. I n the problems treated
earlier in Chapter 4 it seems doubtful that type E regions exist (in terms of
162
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Type D and E Regions 163
Lehmann's development, H is not left invariant by many transform ations ). We
introduce here two possible optimality criteria in the same sprit as the type
D and E criteria which will always be fulfilled by some test under minimum
regularity assumptions. Let
A ( n ) = max A^(TJ) . (7.1)
D e f i n i t i o n 7 .3 . T y p e D A test . A test d>' is of type D A if
maxi
A( V ) - A^(77)] - min ma x[A (r, ) - A*(r/) (7.2)
D e f i n i t i o n 7 .4 . T y p e D M test . A test d<" is of type DM if
m a x ^ M / A d - f a ) ] = min max[A(rj)/A,, , (j))] . (7.3)
These two criteria resemble stringency and regret criteria employed elsewhere
in statistics; the subscripts "A" and " M " stand for "additive" and "multip lica
tive" regret principles. T h e possession of these properties is invariant under
the product of any transformations on 9 (acting tr ivia lly on H ) of the same
general type as those for which type D regions retain their property, and an
arbitrary 1-1 transformation on H (acting triv ial ly on 0 ) , but, of course, not
under more general transformations on ft. Obviously, a type E test automat
ically satisfies these weaker criteria.
Let us now assum e that a testing problem is invariant under a group of
transformations G for which the Hunt-Stein theorem holds and which acts
trivially on 0; that is
g(0,n) = (6,gr,), 9 e G . (7.4)
If d>g is the test function, defined by 4>g{x) = d>(gx), then a trivial compulation
shows that
A tg( V ) =/Xpigv) • (7.5)
It is easy to verify that A^3
( ) j ) = A^gv) implies A(TJ) = A(g-n). Furthermore,if d> is better than $ m the sense of either of the above criteria, then d>g is
better than d>'g. Al l of the requirements of Le hm an n (1959) are easily seen to
be satisfied, so that we can conclude that there is an almost invariant (hence,
in our problems, invariant) test which is of type DA or D ^• This differs from
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164 Group Invariance in Statistical Inference
the way in which invariance is used in page 883 of Lehmann (1959), where it
is used to reduce Q rather than H here.
If the group G is transitive on H, then A( JJ ) is constant as is A^(TJ) for an
invariant <p, which we therefore write simply as A ^ . In this case we conc lude
that if 0 ' is invariant and if <p* of type D among invariant d> ( i.e. i f A ^ .
maximizes A ^ over all invariant <p), then d>" is of type DA or DM among all a).
To verify these optimality properties we need the following lemma.
L e m m a 7.1. Let L be a class of non-negative definite sym me tr ic matrices
of order m and suppose J is a fixed nousingular member of L. If tr J~ L B
is maximized (over B in L) by B = J , then det B is also maximized by J.
Conversely, if L is convex and J maximizes det B, the tr J~ L B is maximized
by B = J.
Proof. Write J~ L B = A. If A = I maximizes tr A, we get (det A) L' M <
tr ~ < 1 = det I . Conversely, if J. maxim ize s det A , it also maxim ize s tr A,
since tr B > tr I implies det(aB + (1 — a)I) > 1 for a sma ll and positive. •
R e m a r k s . The usefulness of this lemma lies in the fact that the generalized
Neyman-Pearson lemma allows us to maximize tr QB^,, for fixed Q more easily
than to maximize A ^ —detB^ among similar level a tests. We can find, for
each Q, a, <pQ which maximizes tr QB^,\ a 4>" which maximizes A ^ is then
obtained by finding a <S> Q for which B$Q — Q~ L .
In problems of the type which we consider here the reduction by invariance
under a group G which is transitive on H often results in a reduced problem
wherein the maximal invariant is a vector R = (Ri,..., Rm)' whose dist ribu
tion depends only on A - (Sy,..., 6 M ) ' where 6\ = B\, $ = ( 0 * , . . . , $ m )' and
such that the density / A of i i wi th respect to a o"-finite measure fi is of theform
where the hi and ay are constants and Q ( r , A ) — o ( ^ £ ; ) as A —> 0 and we
can differentiate under the integral sign to obtain, for invariant test <p of level
m
(7.6)
Til
8T (0 TV ) = all + $ > hA + 5 > £ a t f / r }- #r) f 0(r) p(dr)
(7.7)
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Type D and E Region) 165
as S - » 0. According to Gir i and Kiefer (1969) this is called the symmetric
reduced regular case ( S RR ) .
T h e o r e m 7.1. In the SRR case, an invariant test i>* of level a is of type
D among invariant <p (and, hence, of type DA and DM among all d>) if and
only if <p* is of the form
^^={1}IF
E « {<}<? (7.8)
where c is a constant and qT 1
= const
Proof . In the S R R case, every invariant <f> has a diagonal B$ whose ith
diagonal entry, by (6.7), is 2[hia + £o{53, ai3 R } 4i(R)}]- By Neyman-Pearson
lemma tr QB$ is maximized over such a <j> by a d>* of the form (6.7). Hence
we get the theorem. •
h^ + EotZj ai 3 Rj<l>'{R)
E x a m p l e 7.1. (T
2
-test) Consider once again. Problem 1 of Chapter 4.Let 0 = A
r l
^2
F- 1
^ where F is the unique member of GT with positive diagonal
elements satisfying F F' = E . Th e n 8 is Euclidean p-space, GT operates
transitively on H = {posi tive definite E } but trivially on © and we have the
SRR case. We thus have (6.7) with h{ = - 1 ,
' 1 , d i > 3 ,
- < 0, if i < j , N - j + l, if i - j .
(7.9)
Hotelling's T 2 test has a power function which depends only 5Z;°̂ > so that,
with the above parametrization for 6, we have BT* a multiple of identity.
Hotelling's critical region is of the form £ r> > c - But, when all are equal,
the critical region corresponding to (6.8) is of the form ^( - / V+ p — 1—2 j ) r , > c,
which is not Hotelling's region if p > 1. Hence we get the following theorem.
T h e o r e m 7.2. For 0 < a < l < p < N , Hotelling's T 2 test is not of type
D among Gr-invariant tests, and hence is not of type DA or DM (nor of type
E) among all tests.
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166 Group Invariance in Statistical Inference
R e m a r k s . T he actual computation of a <j>* of type D appears difficult due
to the fact that we need to evaluate an integral of the form
(7.10)
for ft = 0 or 1 for various choices of the Ci'fl and c. Wh en a is close to zero
or 1, we can carry out approximate computations as follows. As a —» 1 the
complement R of the critical region becomes a simplex with one corner at 0.
When p = 2, if we write p = 1 - a and consider critical regions of level a of the
form byi + y 2 > c where 0 < L _ I < b < L , L being fixed but large (this keeps R
close to the origin), we get from (6.10) that p = E a{l-(p{R)) = ( A r - 2 ) c / 2 6 1 / 2 +
o(c) as C - 0. Similarly, E a
{(l - <p(R)Ri} = {N - 2 ) c3
61
-s
'2
/ 8 + o(c) = p 2 bi- 3' 2 /2(N-2) + o(p) as p — 0 while £ 0 ( # , ) = l/N. From (5.18) we obtain
for the power near He
1 - Np + o(p)
2{N-2)b?+ <52
1 - p + o(p) (N - l)pb% + o(p)
2(N - 2 ) 6 * 2(N-2)
+where o{p) and o(J^6i) terms are uniform in A and p, respectively. The
product A ^ of the coefficients of 6\ and b% is maximized when b = (N +
1)/{JV - 1) + o ( l ) , as p —* 0; with more care, one can obtain further terms in
an expansion in p for the type D choice of b. The argument is completed by
showing that 6 < L _ 1 implies that R lies in a strip so close to the ri -axis as
to make EQ (1 - <p(R))Ri too large and E 0{(1 - d>(R))R 2 } too small to yield a
d> as good as that with b = (N + l)/(N - 1), with a similar argument if b > L .
When p is very close to 0, all choices of 6 > 0 give substantially the same power,
a + (Si + £ 2 ) / 2 + o(p 2 ), so that the relative departure from being type D, of
Hotelling's T 2 test or any other critical region of the form bR\ + R 2 > c, with
b fixed and positive, approaches 0 as a -» 1. However we do not know how
great the departure of A r j from A can be for arbitrary a. We can similarly
treat the case p > 2 and also the case a —* 0.
One can treat similarly other problems considered in Chapter 4.
References
1. N. Giri, and J. Kiefer, Local and Asymptotic Minimax Properties of Multivariate
Tests, Ann. Math. Statist. 35, 21-35 (1964).
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Type D and B Regions 167
2. S. L . Isaacson, On the Theory of Unbiased Tests of Simple Statistical Hypothesis
Specifying the Values of Two or more Parameters, A n n . Math. Statist . 22, 217¬
234 (1951).
3. J . Kiefer, On the Nonrandomized Optimality and Randomized Nonoptimality of
Symmetrical Designs, A n n . Math. Statist . 29 , 675-699 (1958).
4. E . L . Lehmann, Optimum Invariant Tests., A nn. Math. Statist . 30, 881-884
(1959).