II--I

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A CLASS OF RANK-ORDER TESTS FOR AGENERAL LINEAR HYPOTHESIS

by

Madan Lal Puri and Pranab Kumar Sen

Courant Institute of Mathematical Sciences,New York University and University of North Carolina

Institute of Statistics Mimeo Series No. 551

September 1967

Research sponsored by the Office of Naval Research,Contract Nonr-285(38), and by the National Instituteof Health, Public Health Service, Grant GM-12868-03.Reproduction in whole or in part is permitted forany purpose of the United States Government.

DEPARTMENT OF BIOSTATISTICS

UNIVERSITY OF NORTH CAROLINA

Chapel Hill, N. C.

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A CLASS OF RANK-ORDER TESTS FOR A

GENERAL LINEAR HYPOTHESIS*

By Madan Lal Puri and Pranab Kumar Sen

Courant Institute of Mathematical Sciences, New York University

and University of North Carolina at Chapel Hill

~~. For a general multivariate linear hypothesis testing

problem a class of permutationally distribution-free rank-order

tests is proposed and studied. Along with a multivariate cum

"multistatistics generalization of Hajek's (1966) results on

the asymptotic normality of rank-order statistics (for

regression alternatives), the asymptotic distribution theory

of the proposed tests and their asymptotic power properties

are studied. Asymptotic optimality of the proposed tests for

specific alternatives is established and a characterization

of the multivariate multisample location problem [cf. Puri

and Sen (1966)] in terms of the proposed linear hypothesis

is also considered.

* Research sponsored by the Office of Naval Research, ContractNonr-285(38), and by the National Institute of Health,Public Health Service, Grant GM-12868-03. Reproductionin whole or in part is permitted for any purpose of theUnited States Government.

2

ratio test is one of the mostly adopted ones. In this paper,

(1958, Chapter 8) and Rao (1965, Chapter 8)]; the likelihood

II-,IIIIII

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= 1, •.• , Nv

l<v<oo,,i < i < N

H : R = OpXq ,o S\. I'IJ

(1.1 )

_ ( (1) (q))1are unknown parameters and ~Vi - c Vi , ••. , cVi ,i

the assumption of multinorma1ity of F(x) is relaxed and aI'IJ,.

class of Chernoff-Savage-Hajek type of rank-order tests is

proposed and studied. These tests are valid for all continuous

where a' = (al , .•• , a p ) and R = ((~jk))j-l p. k-l q1"\1 s\:, - , ••• " - , ••• ,

against the set of alternatives that ~ is non-null.

A variety of tests for Ho in (1.2), based on the assumed

normality of F(x), is available in tne literature [cf. AndersonI'IJ

1. Introduction. Let ue consider a sequence of stochastic"V'VVVVV'V~

are known non-stochastic vectors. Having observed ~v and

assuming some conditions on ~Vi's (to be stated in Section 2),

we want to test the null hypothesis

matrices ~v = (~Vl'···' ~VN ), 1 ~ v < 00, wherev

- ( (1) x(p))' i - 1~vi - XVi ,.~., vi ' - , ... , Nv are independent stochastic

vectors having continuous cumulative distribution functions

(cdf) FVi(~)' ~ € RP, i = 1, ... , Nv , respectively. It is

assumed that

(1.2)

is studied.

rank-order statistics are defined. Section 3 is concerned

3

for i-I < t i i 1, ... , Nv ;Nv< - , =-Nv

j = 1, ... , p .~v . (t ), J

F(x). In Section 2, along with the preliminary notions, these'"

the last Section, relationship of the multivariate multisample

location problem with the linear hypothesis in (1.1) and (1.2)

with certain permutational invariance structure of the joint

utilized for the construction of permutationally distribution­

free tests for Ho in (1.2). In Section 4, by a generalization

of Hajek's (1966) results, the asymptotic joint normality of

the proposed rank-order statistics (for nearby alternatives)

is established. Section 5 deals with the asymptotic power

distribution of Ev (when H in (1.2) holds), and this is then'" 0

properties of the proposed tests for nearby alternatives,

and Section 6 is concerned with the asymptotic optimality of

the proposed tests, granted certain conditions on F(x). In"-

2. Preliminarv notions and fundamental assumDtions. Let Rv(~)~ I'V'V'VV"V'\AI 'V'V\.I~'VVVV'V~ 1

(.) (.) (.)be the rank of Xvr among the Nv observations Xvi , ... , Xv~ ,

vfor i = 1, ... , N ; by virtue of the assumed continuity of F(x)v "-ties may be ignored, in probability, for all j = 1, •.. , p.

Let ~v .(t) be a function defined on (0,1) and taking constant, J

values over intervals [i/Nv ' (i+l)/Nv )' 0 ~ i ~ Nv ' i.e.,

there is some function ~j(t) defined for 0 < t < 1, for which

(2.1)

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definitions lead to asymptotically equivalent results and the

theory follows on almost parallel lines, for simplicity of

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4

for all

:::l, •.• ,p;for j

j =1, ... , p.

J'l

lim [~V .(t) - ~j(t)J2dt ::: 0V-><D 0 ' J

NV

-> 00 as v -> co ), and

(2.4)( II)

j =:: 1, ... , p ,

k=::l, •.• ,q.

Our proposed test for Ho in (1.2) is based on the stochastic

matrix

Consider now

Define

(2.2)

notations and for intended brevity of discussion, we will

consider only the scores defined by (2.2).) Concerning

~V .'s, we assume that, J

(2.3)(I) ~i~oo ~V,j(t) ::: ~j(t) exists for all 0 ~ t ~ 1 and

constant, for j = 1, ... , p, (where of course

where UV(l) ~ ..• ~ UV(N ) denote an ordered sample of Nvvobservations from the ~ectangular distribution over (0,1).

(We may also definea~j)(a) =:: ~j(a/Nv+l) for a =:: 1, ... , Nv

and j = 1, ... , p. However, as it is known that the two

5

(this can always be done by suitably adjusting ~ in (1.1)),

Concerning ~Vils, the following assumptions are made:

k = 1,.,., q;

for all

1- Nv

NV

Sup (2:: [c (k) ]2} < co for all k = 1, ..• , q •vi=l vi

M = ((nz o"))j 0' 1 •""v v,JJ ,J = , ... ,p

NV (k)L cVi = 0 for k = 1, ... , q and 1 < v < co ,

i=l

lim (Max [c (k) ]2j;V [c (k)]2 ] ~ 0V->OO. 1<i <N ' vi 1=1 vi

--v

(2.9) (V)

Let then £v = ((CV,kk'))k,kl=l, ... ,q where

(2.7) (III)

(2.8)(IV)

(2.6)

(2.l0)(VI) Rank of £v = q ~ 1 .

(In fact, by reparameterization in (1.1), £v can always be

made of full rank. Hence, (2.11) is no restriction to our

model.) Let us also define

N

(2 12) 1 (~a(j)(R(j))a(j')(R(jl)• ~V,jjl = Nv-i f;r--V vi --v vi

(2.13)

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(Later on, we shall consider a theorem which establishes that

under certain conditions on F(x), (2.14) holds.) Finally, let.....

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6

j=l, ... ,p;

j=l, ..• ,p,

Rank of ~v = pq .

Rank of M = P > 1 •.....v -

J 1 2 T2~jj = 0 ~j(t)dt - ~j ,

and its rationality will be made clear in the next section.

denoted by H[j](X) (j = 1, ... , p) and the bivariate (joint)

marginals bYH[j,j'](X'y) for j I j' = 1, ... , p. Further, let

-1 (( ))-1 (( jk j'k'We denote by £v = dv, jk; j I k' = dv ' )) the

reciprocal matrix of £v. We may define formally our proposed

test statistic as

Before that, we would like to introduce the following notations.

Let H(x) be a p-variate cdf whose univariate margina1s are.....

(2.l4)(VII)

(where ® refers to the Kronecker-product of two matrices).

By (2.9) and (2.14), we obtain that

(2.18)

(2.16)

7

1, •.. , P ,j f. jl

r (H) = lim'" v->oo

and

Permutation distribution of Sv and the rationalitv of ~v'"V'V'VVVV'VV'V 'V'V"VVVV'VVVV """ """'" """""~ """

Finally, let

It follows from (2.9), (2.11) and (2.11) that under the

assumption (2.23) (VIII) k(H) is of rank p and is finite,

~(H) will also be of rank pq, and ~(H) is finite, provided

lim £v exists. However, SUPvllr~(H) II will be finite, by (2.9).v->oo '"

Under Ho in (1.2), ~v is composed of Nv independent and identically

distributed random vectors. Hence, the joint distribution

of ~v remains invariant under the Nvl permutation of the Nv

vectors ~Vl"'" ~VN among themselves when (1.2) holds. Wev

consider now the basic rank-permutation principle, which is

essentially due to Chatterjee and Sen (1966) and discussed also

by Puri and Sen (1966). We define a p X Nv rank matrix ~v

3.

(2 . 20 ) Tl j j I (H)

(2.21)

(2.22)

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8

R(l) . R(l)vI VNv

(3.1) ~v = = (~Vl' ..• , R ) ,",-VNv. .R(P) R(P)

vI VNv

where ~Vi(1 ) R(P)), for i 1, ..• , Nv · Each= (RVi , ••. , = rowvi

of ~V is a permutation of the numbers 1, ••• , Nv ' there being

thus in all (Nv!)P possible realizations of ~v. Let us

rearrange the columns of ~v in such a way that the first row

has elements 1, ••. , Nv in the natural order, and denote the

*corresponding matrix by ~v. ~v is said to be permutationally

equivalent to a matrix ~~l), if it is possible to obtain

~~l) from ~v only by permutations of the columns of the latter.

* *Thus, corresponding to each ~v' there will be a set ~(~v) of

Nv! possible realizations of ~v' such that any member of this

*set will be permutationally equivalent to ~v. Now, the probability

distribution of Bv over the (Nv!)P possible realizations will

depend on F(~), even when Ho in (1.2) holds (unless F(~) has

coordinate-wise independent marginals). Thus, rank-statistics,

like Qv in (2.6), are, in general, not distribution-free

*under H in (1.2). However, given a particular set ~(Rv)o "'-(of Nv! realizations), the conditional distribution of Bv

*over the Nv! over the Nv! permutations of the columns of Bv

would be uniform under Ho in (1.2), i.e.,

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7

1, ... , P ;

~(H)'"

j 1= j'

limv->co

and

Permutation distribution of Sv and the rationalitv of ~v."V'V'VVV'V'VVV~ IV" t'\I'V" t'\I'V"~ IV"

Finally, let

It follows from (2.9), (2.11) and (2.11) that under the

assumption (2.23) (VIII) k(H) is of rank p and is finite,

~(H) will also be of rank pq, and ~(H) is finite, provided

lim £v exists. However, sUPvlll~(H) II will be finite, by (2.9).v->oo '" .

Under Ho in (1.2), ~v is composed of Nv independent and identically

distributed random vectors. Hence, the joint distribution

of ~v remains invariant under the Nvl permutation of the Nv

vectors ~Vl' •.• ' ~VN among themselves when (1.2) holds. Wev

consider now the basic rank-permutation principle, which is

essentially due to Chatterjee and Sen (1966) and discussed also

by Puri and Sen (1966). We define a p X Nv rank matrix Bv

(2 . 20 ) Tl j j , (H)

(2.21)

3.

(2.22)

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8 II-•IIIIII

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R(l) . R(l)vI VNv

(3.1) ~v = = .(~Vl' .•• , R ) ,",VNv. .R(P) R(P)

vI VNv

where ~Vi(1 ) R(P)), for i 1, .•• , Nv ' Each= (RVi , ... , = rowvi

of ~v is a permutation of the numbers 1, ... , Nv' there being

thus in all (Nv!)P possible realizations of ~v. Let us

rearrange the .columns of ~v in such a way that the first row

has elements 1, ... , Nv in the natural order, and denote the

*corresponding matrix by ~v. ~v is said to be permutationally

equivalent to a matrix ~~l), if it is possible to obtain

~~l) from ~v only by permutations of the columns of the latter.

* *Thus, corresponding to each ~v' there will be a set ~(~v) of

Nv ! possible realizations of ~v' such that any member of this

*set will be permutationally equivalent to ~v. Now, the probability

distribution of ~v over the (Nv!)P possible realizations will

depend on F(~), even when Ho in (1.2) holds (unless F(~) has

coordinate-wise independent marginals). Thus, rank-statistics,

like ~v in (2.6), are, in general, not distribution-free

*under H in (1.2). However, given a particular set ~(Rv)o '"

(of Nvl realizations), the conditional distribution of ~v

*over theNv ! over the Nv! permutations of the columns of Bv

would be uniform under Ho in (1.2), i.e.,

9

may work with a positive-semidefinite quadratic form in the pq

elements of S , where the discriminant of the quadratic form""y

E( S I~} == opxq .""y y ""

whatever be F(~). Let us denote by~y the permutational

(conditional) probability measure generated by the conditional

law in (3.2). Then, by routine computations (along the lines

of Puri and Sen (1966)), we obtain that

where d ok" 'k I I S are defined by (2.15).y,J ,JSince ~y is a stochastic matrix, for actual test con-

struction it is more convenient to use a real valued function

E( Sy ,jk Sy, j 'k' I~} = dy , jkj j 'k I for j, j I = 1, , p,

k,k' = 1, , q,

of ~y as a test-statistics. By an adaption of the same arguments

as in Chatterjee and Sen (1966) and Puri and Sen (1966), we

is the inverse of the matrix Qy' which has the elements

dY,jkjj'k'

in (3.2). This leads to the test statistic Ly '

defined by (2.17). Ly will be stochastically large if ~y is

stochastically different from O. For small values of y"" *(i.e., Ny)' the conditional distribution of Ly ' given ~(Ry)'

can be obtained with the aid of (3.2), and a conditionally

distribution-free test for Ho in (1.2) based on Ly can be

constructed. This, however, requires the evaluation of the

Ny! realizations of ~y (under ~), while ~y is ~-invariant.

The task becomes prohibitively laborious and for large values

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10

Proof: The computation of the means and covariances of

we consider the following.

theorem is a consequence of the first part and the results in

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The proof that M - Ad L> Opxp as v -> (X)"",v ~'"V "'"

Proof:

of v, we are forced to consider the following large sample

THEOREM 3.2. Under~, the joint distribution of the

elements of ~v converges, in probability, to ~ multivariate

normal distribution with null means and covariances given by

approach in which we approximate the true permutation distri-

the elements of ~v follows directly from (3.3) and (3.4).

Since £v is positive definite, in probability (by the preceding

theorem), the asymptotic multinormality of the permutation

brevity, the details are omitted. The second part of the

as v -> co .

follows more or less on the same line as in the proof of

Theorem 4.2 of Puri and Sen (1966), and hence, for intended

bution of ~v (under ~v) by that of a chi-square distribution

with pq degrees of freedom (d.f.). As a basis of this study,

(3.4) .

(2.9) through (2.23). Q.E.D.

NvTHEOREM 3.1. Let Hv(~) = (l/Nv ) f=l FVi(~)' and let

.f6v = lI.(H) IH == H be defined ~ in (2.21). Then, under the

assumptions (I)vto (V) of Section 2, [~v _ ~] ~> £px p ~

V -> co. Thus, under the assumptions (VI) and (VII), ~v

is positive definite, in probability, and £v - ~(Hv) JL> opq~pq"'" .,

11

By virtue of the preceding theorem, we arrive at the

if ~V

distribution of S follows readily by an appeal to the multi­""v

> 2 , reject H i (1 2)v n • ;- Apq,a 0

Hence we have the following large sample test procedure:

< X~q,a ' accept Ho '

variate extension of the well-known Wald-Wolfowity-Noether­

Hoeffding-Motoo-Hajek permutational central limit theorems

[cf. H~jek (1961, p. 522)]. Hence, the theorem.

following theorem through a few simple steps.

2 2where P(Xt ~ Xt,a} = a(O < a < 1), the level of significance.

4.~~ 2£ ~v· We introduce the folloWing

notations. Let

and let HV[j](X) be the marginal of the jth variate of Hv(~),

THEOREM 3.3. Under the conditions (I) to (VIII) of

Section 2, ~v' defined by (2.17), has ~ permutation distribution

(under~) asymptotically converging, in probability, to a

chi-square distribution~ pq degrees of freedom.

(4.1)

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12

for j, j' = 1, .•• , P and k, k' = 1, •.• , q; the corresponding

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, I

• ~j(HV[j]~X))~j(HV[j](Y))dFvr[j](X)dFvs[jJ(Y)

+ rr FVi [ ,](X)[l-FVi[j](Y)]J,J -00 <X <Y<oo J

Ajj(i ) =jl"f FVi[j](X)[l-Fvi [ '](Y);rs J -co <X <Y<OO J

. Ajjl(i;rs) ,

, I

. ~ j ( Hv [ j ] (x ) ) ~ j I (Hv [ j I ] ( y) ) dFvr [ j ] dFV s [ j I ] (y) ,

pqxpq matrix is denoted by ~v. Throughout this section it will

for j = 1, ... , p. Also, let FVi[j](x) and FVi[j,j'](x,y)

be the univariate (jth) and bivariate ((j,j' )th) marginals

corresponding to FVi(~)' for j I j' = 1, .•• , p. We define

for j I jl = 1, ... , P and i, r, s = 1, ..• , Nv; the subscript

V in (4.2) and (4.3) is understood. Let then

for i, r, s = 1, ... , Nv and j = 1, .•. , p;

(4.2)

(4.4)

rOO .00

(4.3) Ajjl(i;rs) = J-ro

)_00 [FVi[j,j,](x,Y)-Fvi[j](X)Fvi[jl](Y)]

as v -> 00, and hence

13

-> 0

rna x [A.. I ( i . ) - 'Tl j . , (F)]} = 0 ,l~j,jl<p JJ ,rs J

( maxl<i<N--v

~v is positive definite and finite.

(Jv, jkj j I k' - Cv , kk I 'Tl j j , (F) -> 0 as v -> 00,

( maxl<i,r,s<N- -v

limv->oo

By virtue of (2.8) and (2.9), maxl<i<N--v

limv->oo

Proof:

(4.8)

Before presenting the main theorem of this section, we consider

having a continuous density function f(x), x € RP , and (ii)"" ""

be assumed that (i) F(x) in (1.1) is absolutely continuous""

the following lemma.

LEMMA 4.1. Under (1.1), (2.8) and (2.9),

2: - A(F)® C -> opq)(pq as v -> 00. Hence under (2.11)""V "" ""v "" -and (2.23), ~v is positive definite in ~ limit as v -> 00.

for all j,jl = 1, ••. , Pj k,k' = 1, ... , q. This completes

where 'Tl •• ,(F) is defined by (2.20). From (4.4), (4.7), (2.8)JJ

and (2.9), we obtain after some simplifications that

the first part of the proof. The second part follows from

From (4.2), (4.3), (4.6) and some routine computations, it

follows that

(4.5)

(4.6)

(4·7)

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results on limits of sequences. Q.E.D.

Let us now introduce the following notations. Let

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for all j,j' = l, ... ,p; k,k' = l, ... ,q,

COV{ZV,jk' ZV,j'k'} = crV,jkjj'kl ,

NV(4.11 ) ZV,jk = f=l ZVi (jk) for j = 1, ... , Pj

k = 1, ... , q.

14

Straight-forward but somewhat lengthy computations yield that

where cr v 'k 'lk,IS are defined by (4.4)., J , J

In this context, we bring now the findings of Hajek

(1966), who considered the special case of p = q = 1 and

for j = 1, •.. , Pj k = 1, ... , q and i = 1, ... , Nv • Finally

let

for j = 1, ... , p, i,i' = 1, ... , Nv . Also let

(4.10 )

(where u(x) is equal to 1 or 0 according as x is positive or not),

(2.11), (2.23) (with H =F), (4.8) and some well-known

(4.12)

that

as Y -> co

o ,2lim E[[Sy 'k-E(Sy 'k)] - Zy 'k} /Var(Zy 'k)v->oo ' J , J , J , J

details, we obtain that

for all j = 1, ••. , p; k = 1, ••• , q. We shall utilize (4.13)

to establish a comparatively stronger result. With this end

in view, we first consider the following lemma.

15

discussed the asymptotic normality of the corresponding rank­

order statistic (which is just anyone of the SY,jk's). Thus

referring to his elegant paper and thereby avoiding the

By virtue of (4.12), (4.13) and lemma 4.2, it follows

LEMMA 4.2. Let ~y = (a yl , ••. , avt ) and £y = (byl '···' byt )

t > 1, be two sequences of stochastic vectors, such that

v(aYj - b ,)/V(a ,) -> 0 as y -> 00 for all j = 1, ..• , t.YJ YJ --

Then

(4.13)

making use of the conditions stated in the lemma.

for all j, £ = 1, .•• , t.

The proof follows by expressing cov(ayj,ay£) as

COV(byj,by£) + Cov(byj,ay£-by £) + cov(aYj-bYj,by£)

+ cov(aYj-bVj,ay£-by£)' applying Cauchy-Schwarz inequality to

the second, third and fourth terms on the right hand side and

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f or all j, j I = 1, ..• , p; k, k I = 1, •.. , q.

16

Thus it follows from (4.13) and Lemmas 4.2 and 4.3 that

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-'I,

p q

Zv = 2:: 2:: £jk ZV,jk 'j=l k=l

N1 ~~ t;q (k) (k) (j)= N L-. JJ'k( c Vi -c Vi )B j (r i) (XVi ) ,V r=l j= k=· '

where

An immediate consequence of Lemma 4.2 is the following,

for i = 1, ... , Nv . By definition, gi{~Vi) are independent

random variables. Further

view, we define

by showing that any arbitrary linear combination of ZV,jk'S

has asymptotically a normal distribution. With this end in

LEMMA 4.3. If ~V has asymptotically a multinormal distri­

bution with~ vector ~V and dispersion matrix kv and if for

each j(= 1, ... , t) v(aVj - bVj)/V(a Vj ) --> 0 as v --> 00, then

bv ~ also asymptotically the~ multinormal law.

for proving the asymptotic normality of ~v' it is sufficient

to show that (ZV,jk' j = 1, .. " p; k = 1, ... , q) has asymptoti­

cally a multinormal distribution. This will be accomplished

where Jjk's are real and finite and not all of them are zero.

From (4.10), (4.11) and (4.15), we obtain that

Nv(4.16) Zv = f=l gi(~Vi) ,

(4.17)

(4.15)

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17

is finite and positive by (4.5). Proceeding then precisely

on the same line as in Htjek (1966, Section 5 specifically and

using a linear combination such as (4.15) with his Yv's),

it can be shown that under (4.5), the random variables

(gi(~Vi)) satisfy the Lindeberg condition for the applicability

of the central limit theorem. Hence, we arrive at the

following.

THEOREM 4.1. Under the assumptions (I) - (V) of Section 2

and (4.5), the elements of ~v - E~v has asymptotically a

multivariate normal distribution with zero means and dispersion

matrix L: v •""

Now, under the model (1.1) and granted the assumptions

made in Section 2, it follows from routine computations that

for all j = 1, ... , p; k = 1, ... , q; and by Lemma 4.1, the

covariance matrix ~v is asymptotically equivalent to k(F)~ £v·

In the next section, we shall make use of this result for studying

the asymptotic properties of the test based on ~v in (2.17).

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18

We denote by

...* f;;..L ~q ~ S S '\')jj t (F) .Ckv

kt~v = > > v 'k V jtk t 'I

= j t =1 = k t =1 ' J ,

Asvmntotic nronerties of the test.~~ 'V'\J ~ 'V'\J'V'\J

(5.4)

5·

has asymptotically (under (1.1) and the assumptions of Section 2)

where

random variables, it follows that

Further, using Theorems 3.1 and 4.1, it is straight-forward

to show that under the assumptions made in Section 2,

,a non-central chi-square distribution with pq degrees of

freedom and non-centrality parameter

for all j,jl = 1, ... , p, where ~jjl IS are defined by (2.19),

(2.20) and (2.21). Then, using Theorem 4.1, Lemma 4.1,

(4.19) and some well-known result on the limiting distributions

of quadratic forms in asymptotically normally distributed

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19

Thus, we arrive at the following theorem .

THEOREM 5.1. Under (1.1) and the assumptions made in

Section 2, the statistic ~v has asymptotically a non-central

chi-square distribution with pq degrees of freedom and non­

centrality parameter ~~, defined ~ (5.3).

Theorem 5.1 may be used to study the asymptotic power

properties of the test in (3.5) and to compare its efficiency

relative to standard parametric tests. Since in general

linear hypothesis testing problems, the different test-criteria

do not have uniformly good or bad performances, [cf. Anderson

(1958, Chapter 8)], it may be quite involved to give a neat

picture of the relative-efficiency of the proposed tests with

respect to all the parametric ones. Among the notable

parametric tests, Roy's largest characteristic root-criterion

is admissible and powerful against certain alternatives,

while the likelihood ratio test is so for certain other

alternatives. However, borrowing the ideas of Wald (1943),

it can be shown that in the sense of Wald, the likelihood

ratio test has asymptotically the best average power (on

suitable ellipsoids in the parameters ~jk's). In this sense,

the likelihood ratio test may be regarded as optimum. As such,

we will consider the asymptotic relative efficiency of our

proposed test with respect to the likelihood ratio test .

1, ... , p

20

Following the notations of Anderson (1958, Chapter 8),

we define the log-likelihood criterion (adjusted by a suitable

multiplying factor) by Tv (say,) and conclude from his Theorem

8.6.2 that Tv has asymptotically (under Ho in (1.2)) a chi­

square distribution with pq degrees of freedom. Hence, the

same test procedure as in (3.5) applies to the likelihood ratio

test (with Zv replaced by Tv). We denote the covariance

matrix (assumed to be finite) of F(x) by r = ((p "k(F)))" k~ ~ sJ J, =

and assume it to be positive definite. The corresponding

reciprocal matrix is denoted by ,-1 = ((~jk(F))). Since the

likelihood ratio-test criterion is exclusively based on linear

estimators of ~, generalizing the method of approach of

Eicker (1963) and following essentially some routine steps it

can be shown that under the assumptions made in Section 2,

Tv has asymptotically a non-central chi-square distribution

with pq degrees of freedom and noncentrality parameter

It is seen that in multivariate situations, the asymptotic

relative efficiency (A.R.E.) of the test based on Zv with

respect to the one based on Tv' as measured by ~Z/~T' depends

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(i) If p = 1, no matter whatever be ~ and £v

for the two sample location problem. A few particular cases

may be worth mentioning here. First, if ~(u) = u (i.e., the

Wilcoxon scores case), (5.7) reduces to 12~11(F)( j' CD f 2 (x)dx)2,-00

which has known values (as well as lower bounds) for various

which happens to coincide with the usual efficiency expressions

matrices and as such

F(x). Secondly, if ~(x) is the inverse of the standard

not only on ((v, ,,(F))) and ((~, ,,(F))) but also on RandJJ . JJ. ~

£v. The following points are worth noting in this context.

for p = 1.

normal cumulative distribution function (i.e., the normal

scores case), (5.7) is always greater than or equal to unity,

the equality sign holds only when F is normal. This clearly

indicates the asymptotic efficiency of the proposed tests

it readily follows that both ~(F) and v(F) are diagonal"" ""

Hence, if we use the normal scores for ~Vj'S, again (5.9)

p(5.8) (ii) If F(~) =TT F[J'](X J')'

j=l

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becomes greater than or equal to unity, where the equality

sign holds only when F(x) is multinormal.

(iii) If we use the normal scores for ~Vj'S, and if the

parent cdf F(x) is a multinormal cdf, it readily follows from"'"

(2.19), (2.20) and (5.1) that ~(F) = ~(F), and hence, from

(5.3), (5.6) that 6~/ 6 T = 1. Or in otherwords, for parent

normal distributions, the use of normal scores leads to

asymptotically most efficient tests.

(iv) In general, for arbitrary F(~), 6 ~/ 6 T is bounded

below and above by the minimum and maximum characteristic

roots of ~(F)~-l(F), (the proof follows by a straight-forward. .

application of a theorem by Courant (cf. [11]) on the bounds

of the ratio of two quadratic forms). Now, the bounds of

£(F)~-l(F) have been studied in detail by the authors (Sen

and Puri (1967)) in connection with the multivariate one sample

location problem. As such, to avoid repetition, the details

are omitted.

6. AsvmDtotic oDtimalitv of ~ for certain F(x). We shall~~"""""""'''''''''''''' v""""""",,,,,,,,,,,~ "'"

study the asymptotic optimality of ~v for a class of parent

distributions. We assume that F(x) has the absolutely continuous"'"

density function f(x), and define,"'"

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for i = 1, ... , P .

I dfi[i](x) = dx f[i](x), for i = 1, ... , P •

We define the statistics

N~ (k) I (j), (1) (p)

UV,jk =~ GVi [fj(XVi ~Vi , ... , ~Vi )/f(~Vi)] ,

Let then

for j = 1, ... , P and k = 1, ..• , q. We also define

reciprocal matrix of !v' and

for all j,jl = 1, ... , p, and let T = ((T'k 'Ik')) be the'" . J,J

pqxpq matrix whose elements are cv,kk' X~jj" where £v is

defined by (2.10). Finally let T-l = ((Tjk,j'k')) be the"'v v

(6.2)

(6.4)

Then by an adaptation of the line of approach of Wald (1943),

*we can show that (i) under Ho in (1.2), Uv has asymptotically

a chi-square distribution with pq degrees of freedom (provided

((~jj')) is finite and positive definite) and (ii) the test

*based on Uv has asymptotically the best average power (on the

family of ellipsoids

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not aware of necessary conditions for the same.

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and consequently,

*to Uv )' will

Let us now define

Nv (k)' (") (")

VV,jk = ~ c Vi fj[j](Xvr )/f[j](Xvr ) ,

( 6.8)

We shall show that under certain conditions on F(x), the test'"

based on ~v has also asymptotically the best average power

on the same family of ellipsoids, provided ~jfS are chosen

suitably. Our treatment deals with a set of sufficient

conditions for this asymptotic optimality and the authors are

for j = 1, ... , p, k = 1, ... , q. If (6.9) holds, it follows

that UV,jk'S are linear functions of VV,jk'S,

the quadratic form based on VV,jk'S (analogous

*also have the same properties as that of Uv .

We now define

(6.11) ~j(u) = f;[j](F[~](U))/f[j](F[3]~u)) , 0 < u < 1,

j = 1, ... , P ,

(6.10)

where h jjl 's are real constants, not all zero, for j = 1, ..• , p.

(6.9) holds for (i) the multivariate normal distribution,

(ii) any coordinate-wise independent distributio~and it may

also hold for many other distributions.

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and define S 'k's as in (2.5). Following then H'jek's (1962)V,Jelegant approach it is seen that S - V converges inv,jk v,jkquadratic mean to zero for all j = 1, ... , p, k = 1, ... , q.

Consequently, on using Lemma 4.2 and some routine computations,

we obtain that for ~j'S given by (6.11), Lv in (2.17) is

stochastically equivalent to the quadratic form in VV,jk'S,

under (1.1) and the assumptions of Section 2. Since, under

*(6.9), this quadratic form in VV,jk'S is also equal to Uv in

(6.7), it follows that under (1.1), the assumptions made in

Section 2 and (6.9), (6.11),

(6.12)

Consequently, we arrive at the following.

THEOREM 6.1. Under (i) the model (1.1), (ii) the- ~

assumptions made in Section 2, and (iii) (6.9) and (6.10)

Lv has asymptotically the best average power on the family

of ellipsoids in ~'k's, defined by (6.8).---=---- - J -

In particular, if F(x) is normal, the condition (6.9)1'\1

.. -is satisfied and (6.11) lead to the coordinate-wise normal

scores. Hence, ~v based on normal scores statistics leads to

asymptotically best test for the family of ellipsoids in

(6.8). A special case of this theorem (that is, for p = 1)

is dealt in an interesting power of Matthes and Truax (1965).

26

7 - A characterization of the multivariate multisamole location'"~ tV'\J IV'V'v~~~

~- Let ~ 1<1' ••• , Xknk

be nk independent and identically

distributed p-variate random variables having a continuous

p-variate cumulative distribution function Fk(~) for k::::l, •.. ,c(>2).

In the multivariate multisample location problem [cf. Puri and

Sen (1966) and the other references cited therein], we may let

Fk(X) = Fl(x - e), K:::: 2, ... , c, e l :::: 0'" '" "'k '" '"

We define N :::: n l + ... + nc ' and consider a sequence of N

p-vectors ~l' .•• ' ~N' of which the first nl observations are

from the first sample, the next n2 from the second sample and

so on. Thus, (7.1) will correspond to (1.1), if we let

q :::: C - 1, ~jk :::: ej,k+l for k = 1, ... , c-l, and where ~ViIS are

null vectors for an 1 < i ~ nl , £Vi = (0, 1, 0, ••• , 0)1

for nl + 1 ~ i ~ nl + n2 , and so on. Thus, the results

derived in this paper, also generalizes the results of Puri

and Sen (1966). The statistic Lv in (2.17) can be shown

to be identical with the corresponding LV in Puri and Sen (1966),

when (7.1) holds. Consequently, the efficiency considerations

made in Sections 5 and 6 also applies to the multisample

multivariate location problem.

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[2]

[3]

[4 ]

[5]

[6]

[8]

REFERENCES

Anderson, T. W. An introduction to multivariate

statistical analysis. New York: John Wiley, 1958.

Chatterjee, S. K., and Sen, P. K. (1966). Nonparametric

tests for the multivariate multisample location problem

S. N. Roy memorial volume. (Edited by R. C. Bose et al).

University of North Carolina (in press).

Chernoff, H., and Savage, I. R. (1958). Asymptotic

normality and efficiency of certain nonparametric test

statistics. Ann. Math. Statist. 29, 972 - 994.

Eicker, F. (1963). Asymptotic normality and consistency

of the least squares estimators for families of linear

regressions. Ann. Math. Statist. 34, 447 - 456.

H'jek, J. (1961). Some extensions of the Wald-Wolfowitz­

Noether theorem. Ann. Math. Statist. 32, 506 - 523.

H{jek, J. (1962). Asymptotically most powerful rank

order tests. Ann. Math. Statist. 33, 1124 - 1147.

H§jek, J. (1966) Asymptotic normality of simple linear

rank statistics under alternatives, I. (to be published).

Matthes, T. K. and Truax, D. R. (1965). Optimal

Invariant Rank Tests for the k-Sample Problem.

Ann. Math. Statist. 36, 1207 - 1222.

Puri, M. L., and Sen, P. K. (1966). On a class of

multivariate multisample rank-order tests. Sankhya,

Ser A. 28, 353 - 376.

[10]

[11]

[12]

Rao, C. R., Linear statistical inference and its

applications. New York: John Wiley, 1965.

Sen, P. K., and Puri, M. L. (1967). On the theory of

rank order tests for location in the multivariate one

sample problem. Ann. Math. Statist. 38, 1216-1228.

Wald, A. (1943). Tests of statistical hypotheses

concerning several parameters when the number of

observations is large. Trans Amer. Math. Soc. 45, 426 ­

482.

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