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-------------.---------------.. ---------------------

ctd. on inside back cover

Lecture Notes in Statistics

Edited by J. Berger, S. Fienberg, J. Gani, K. Krickeberg, and B. Singer

58

Ole E. Barndarff-Nielsen Preben Blresild Paul Svante Eriksen

Decomposition and Invariance of Measures, and Statistical Transformation Models

Spri nger-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong

Authors

Ole E. Barndorff-Nielsen Preben BI<Esild Department of Theoretical Statistics Institute of Mathematics, Aarhus University 8000 Aarhus, Denmark

Poul Svante Eriksen Department of Mathematics and Computer Science Institute of Electronic Systems, Aalborg University Center Strandvejen 19,9000 Aalborg, Denmark

Mathematical Subject Classification: 20-02, 20G99, 22-02, 22D99, 22E99, 28-02, 28A50, 28C 10, 53C21, 53C65, 57S20, 57S25, 58A 15, 58C35, 62-02, 62A05, 62A 10, 62E 15, 62F99, 62H99.

ISBN-13: 978-0-387-97131-5 e-ISBN-13: 978-1-4612-3682-5 001: 10.1007/978-1 -4612-3682-5

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re·use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer·Verlag Berlin Heidelberg 1989

2847/3140·543210 - Printed on acid·free paper

Preface

The present set of notes grew out of our interest in the study of

statistical transformation models, in particular exponential transfor­

mation models. The latter class comprises as special cases all fully

tractable models for mUltivariate normal observations. The theory of

decomposition and invariance of measures provides essential tools for

the study of transformation models. While the major aspects of that

theory are treated in a number of mathematical monographs, mostly as

part of much broader contexts, we have found no single account in the

literature which is sufficiently comprehensive for statistical pur­

poses. This volume aims to fill the gap and to indicate the usefulness

of measure decomposition and invariance theory for the methodology of

statistical transformation models.

In the course of the work with these notes we have benefitted much

from discussions with steen Arne Andersson, J0rgen Hoffmann-J0rgensen

and J0rgen Granfeldt Petersen. We are also very indebted to Jette Ham­

borg and Oddbj0rg Wethelund for their eminent secretarial assistance.

May 1989

Ole E. Barndorff-Nielsen

Preben BI~sild

Poul Svante Eriksen

CONTENTS

Preface

1. Introduction

2. Topological groups and actions

3. Matrix Lie groups

4. Invariant, relatively invariant, and quasi-invariant

measures

5. Decomposition and factorization of measures

6. Construction of invariant measures

7. Exterior calculus

8. Statistical transformation models

Further results and exercises

References, with author index

Subject index

Notation index

Page

1

2

15

28

41

53

66

74

113

133

137

145

1. Introduction

Decomposition or disintegration of measures and construction of

invariant measures play essential roles in mathematics and in various

fields of applied mathematics.

In particular, the mathematical methodology in question is required

for certain advanced parts of parametric statistics. See, for instance,

Fraser (1979), Muirhead (1982), Barndorff-Nielsen, Bl~sild, Jensen and

J0rgensen (1982), Eaton (1983), Baddeley (1983), Andersson, Br0ns and

Jensen (1983), Farrell (1985), and Barndorff-Nielsen (1983, 1988).

However, a comprehensive exposition of the relevant mathematical

results is not available in the statistical literature, nor in the

mathematical, and where the various results can be found they are often

rather inaccessible, being included in some advanced and more compre­

hensive mathematical treatise.

The present notes constitute an attempt to remedy this situation

somewhat, particularly as concerns the need in statistics. An important

starting base for the work has been an excellent set of notes by An­

dersson (1978). We have tried to strike a suitable balance between a

complete account of the mathematical theory and a mere skeleton of

formulas, thus providing the interested reader with enough details and

references to enable him to complete the account mathematically, if he

so desires. The reader is assumed to have some, rather limited, elemen­

tal knowledge of topology, group theory and differential geometry, and

- if he or she wishes to study the statistical applications in section

8 - a considerable knowledge of parametric statistics.

In section 2 we discuss the actions of groups on spaces, with the

ensuing concepts of orbital decompositions and right and left factori­

zations of groups. Some basic results on matrix Lie groups and the asso­

ciated Lie algebras are provided in section 3. section 4 contains the

definitions of invariant, relatively invariant and quasi-invariant

measures and conditions for the existence of such measures, while

methods for constructing invariant measures are considered in section

6. Part of the material of section 6 is intimately connected to ques­

tions of decomposition and factorization of measures which is the sub­

ject of section 5. The exterior calculus of differential geometry often

provides an efficient and elegant way of decomposing a measure or find­

ing an invariant measure. The most relevant aspects of exterior calcu­

lus are outlined in section 7.

The final section 8 illustrates the usefulness of the mathematical

tools by deriving the key properties of statistical transformation

2

models. That section has been organized with the aim of enabling a

reader with only a limited knowledge of the material in sections 2-7 to

follow the main lines of the developments. However, a good general

knowledge of parametric statistics is needed for a full appreciation of

the discussion in section 8.

Another area of statistics in which decomposition, or disintegra­

tion, of measures is of great importance is that of spatial statistics.

However, except for a derivation of the Blaschke-Petkantschin formula,

we do not touch upon that area here. In particular, we do not discuss

Palm measures and Gibbs kernels. outstanding accounts of the mathemati­

cal theory of these are available in Kallenberg (1983) and Karr (1986).

See also stoyan, Kendall and Mecke (1987) and Karr (1988).

Examples are considered throughout the book, and a small collection

of further results and exercises is included at the end. In particular,

an outline of the main properties of exponential transformation models

is given as exercise 25. (Most of the examples in section 8 are, in

fact, concerned with models of this type.)

Some of the results discussed, especially in section 8, are novel

but most of the material stems from the existing literature. The rela­

tion of the material presented here to other parts of the literature is

indicated in the bibliographical notes which conclude each of the sec­

tions 2-8.

2. Topological groups and actions

In this section we introduce the concept of an action of a topologi­

cal group G on a topological space ~. Furthermore we lay down a set

of topological conditions to ensure that there exists a measure on ~

which is invariant under the action of G.

A group G that is also a topological space is called a topological

group if the mapping

G x G ~ G

is continuous (with respect to the product topology on G x G and the

topology on G).

Let ~ be a topological space and G

ping ~ from G into the symmetric group

a topological group. A map­

~(~) over ~, i.e. the

3

set of one-to-one transformations of ~ onto ~ with composition of

transformations as composition rule, is called an action of G on ~

if

(i) ~ is a homomorphism

(ii) the mapping

(g,x) .... ~ (g) (x)

is continuous.

It is clear from (ii) that ~ (g) is continuous and since (i) im-

plies that -1 -1 ~(g) = ~(g ), we must have that ~ (g) is a homeomor-

phism for all 9 € G. For short we often write gx for ~ (g) (x) . The

subset Gx = {gxlg € G} is called the orbit of x, and the set ~ is

partitioned into the collection of orbits. We will call this collection

the orbit space and denote it by G\~. The mapping

'IT: ~ .... G\~

x .... Gx

is called the orbit projection. We will assume that G\~ is endowed

with the quotient topology, i.e. A ~ G\~ is open if 'IT-1 (A} ~ ~ is

open.

We say that G acts transitively on ~ if there is one orbit only,

i.e. for every x 1 ,x2 € ~ there exists a 9 € G such that gX1 x 2 .

Since ~ is said to be a homogeneous space if for all x 1 ,x2 € ~

there exists a homeomorphism f of ~ so that f(X 1 ) = x 2 ' it is

clear that if G acts transitively on ~ then ~ is a homogeneous

space.

Example 2.1. Linear and affine groups. Consider the general linear

group GL(n) consisting of the invertible n x n matrices endowed

with the usual matrix multiplication. The group GL(n) is a topologi­

cal group acting linearly on ffin by

GL(n) x ffin .... ffin

(A,x) .... Ax

4

where Ax is the ordinary matrix-vector product. The space rnn is

divided into the two orbits {OJ and rnn\{O}, so that GL(n) acts

transitively on rnn\{O}.

The subgroup GL(n) consisting of matrices A having positive

determinant is denoted by GL+(n) and is termed the positive general

linear group.

The group GL(n) is a subgroup of the general affine group GA(n)

which consists of all pairs [A,f] where A € GL(n) and f € rnn, the

group operation being given by

[A' ,f/] [A,f] [A' A,A' f+f ']

Restricting A to have positive determinant one obtains the positive

general affine group GA+(n).

The two groups GA(n) and GA+(n) act transitively on rnn by

([A,f],x) -+ Ax + f . o

The action of G on ~ is said to be free if all orbits are

'copies' of G, i.e. for all x E ~ we have that, if y E Gx then

there exists only one g € G such that y = gx.

For x E ~ we define Gx ' the isotropy group or isotropic group of

x, as the subgroup of G consisting of the elements that leave x

fixed, i.e. Gx = {g E Glgx = x}. Obviously, G acts freely if and

only if Gx {e}, the identity element, for all x E ~.

Now let us choose orbit representatives u, i.e. on each orbit of

~ we select one point u to represent that orbit. We may think of u

as a function of x, u(x) being the representative of the orbit GX,

and this function is a maximal invariant since u(gx) u(x) and

u(x1 ) ~ U(X2 ) if x2 (GX1 . If G acts freely then there exists a

unique element z(x) E G such that x = z(x)u(x), and since z(gx)

gz(x) we have that z is eguivariant. In this way we have defined a

kind of coordinate system on ~ and the mapping x -+ (z,u) is termed

an orbital decomposition of x.

Example 2.2. Scale action.

acts on rnn by

The positive multiplicative group

5

* n n IR+ x IR ~ IR

(a,x) ~ax.

The orbits are {O) and half-lines extending from zero. One finds that * n -1 IR+ acts freely on IR \{O} and that x ~ (lIxll,lIxll x), II II indicat-

ing the Euclidean norm on is an orbital decomposition of x.

* Example 2.3. Location-scale action. Let G = GA+(1) = IR+ x IR be

the group of location-scale transformations. Then, given an Xo €

IR~{O}, G acts on IR n by

G x IR n ~ IR n

([a,f],x) ~ ax + fxO .

The isotropic groups are seen to take the form

{{[a,a(1-a)]\a > O}

{e}

if

otherwise.

a € IR

It follows that G acts freely on ~ = IRn\{x\x

<.,.> be an inner product on IR n and define

-x -1 <xO,xo> <xO,x>

-1 -2 <xO,xo> <x,x> - X

axO' a € IR}. Let

Then -1 -u(x) = s(x) (x-xxo) is an orbit representative and x ~

«s(x),x),u(x» is an orbital decomposition. o

o

Example 2.4. Commutator action. Let H = {V i }1=1 be a finite sub­

group of the orthogonal group O(n) = {V € GL(n) \UU* = In}. Consider

the commutator group

C(n,H) {A € GL(n) \AV i ViA, i=1, ... ,p}.

Then C(n,H) acts on IR n by

6

(A,x) -+ Ax.

The structure of such actions is totally clarified and, in a statisti­

cal context, the structure is described in Andersson, Br0ns and Jensen

(1982). Andersson, Br0ns and Jensen study the class of n-dimensional

normal distributions with a covariance matrix which is invariant under

* H. The set of such covariance matrices is given by (AA IA E C(n,H)}.

Here we just mention a few properties of the prescribed actions.

Define the n x n symmetric matrix

s (x)

and suppose

~ = (x E mnls(x) is positive definite}

is non-empty. Then it can be shown that ~ is open and mn\~ has

Lebesgue measure zero. Moreover, C(n,H) acts freely on ~.

Further, under the class of n-dimensional normal distributions with

mean vector

we have that

o and a covariance matrix of the form * AA ,

sex) is the maximum likelihood estimate of

A E C(n,H),

* AA •

A very simple example is provided by taking H to be the group of

permutations, i.e. if a E ~(n) - the symmetric group of order n

then U(a) E H is the mapping

and

C(n,H)

Furthermore,

s(x). . 1,J

{A {aij } € GL(n) I aii a,

i=j

i=1, ... ,n, a .. =b, 1J

i;o!j } .

7

The associated normal model is the class of distributions that are

invariant under permutation of the coordinates. o

When G does not act freely, it is often the case that we can

choose an orbit representative u so that the isotropic group of u

is independent of u, Le. Gu(x) = K for all X E ~. In this case

there exists for any pairs x and x' of elements of ~ a one-to-one

mapping of Gx onto Gx ' and we say that ~ has orbits of constant

~. Furthermore, there exists a unique element z(x) E G/K

{gKlg E G}, where gK = {gklk E K}, such that x = gu(x) for

g E z(x). In this case the mapping x ~ (z,u) is also termed an orbi­

tal decomposition.

Example 2.5. The orthogonal group O(n) = (U E GL(n) luu* = In}

acts as a subgroup of GL(n) linearly on

collection of balls (S(r) Ir > O}, where

we choose the orbit representatives u as

it is clear that the isotropic group K of

trices

{~ ~}, U E O(n-l).

mn\{O}. The orbits are the

S(r) = {x E mnlllxll = r}. If * n u(x) = (O, ... ,O,lIxll) Em,

u(x) is the set of ma-

This shows that mn\{O} has constant orbit type. o

Example 2.6. Let SOl(l,q) denote the connected component of the

pseudo-orthogonal group O(l,q) which contains the identity element

e. Here O(l,q) is the group of linear transformations of mq+1

leaving the symmetric form x~-x;- ... -x~+l invariant, and sol (l,q)

constitutes a subgroup of O(l,q). If we consider the (q+l) x (q+l)

matrix

I l,q

then, as will be discussed in example 3.2, sol (l,q) has the matrix

representation

8

(A € GL(q+1) Idet A 1, * AIl A = I1 }. ,q ,q

As a subgroup of GL(q+1), sol (l,q) acts linearly on mq+1 and the

orbit of u~ = (1,0, ... ,0) € mq+1 is the unit hyperboloid

1}

where the scalar product * is defined by x*y The set

{O} constitutes an orbit of a particular type. For q > 1 and

x € mq+1,{0} we obtain different orbit types according to where x*x >

0, x*x = 0 or x*x < O. If x*x ~ 0 we define the 'hyperbolic

length' as r(x) = Jx*x. We are in particular interested in the open

subset

* if and only if If u(x) r(x)uo € ~, X € ~, then A € G u(x)

A {~ g}

where U is an element of GL(q) such that UU * = I and det U 1. q

(We use det to indicate the determinant of a matrix and I I to

indicate the numerical value of the determinant of a matrix A, i.e.

IAI is short for Idet AI.) The set of such matrices u constitutes

the special orthogonal group SO(q), cf. example 3.2. Thus, for

we have that Gu(x) is essentially equal to SO(q) and we write

X € ~

GU(X) = SO(q). since Gu(x) does not depend on x we have that

is of constant orbit type.

The above-mentioned action on the orbit corresponding to r(x)

1, has been studied in a statistical context by Jensen (1981), cf.

also example 8.7. o

Example 2.7. Consider the vector space gl(p,n) of p x n ma­

trices. The group G = O(p) x O(n) acts on gl(p,n) by

G x gl(p,n) ~ gl(p,n)

* «U,V),X) ~ uxv

9

Suppose p:: n i=l, ... ,p where 2 2 A1 (X» ... >A (X) - - p

are the ordered eigenvalues of * XX .

u(X) = (diag (A 1 (x), ... ,Ap (X», !V

Then

is an orbit representative. The orbit type of X is determined by the

set of equalities in the relation A1(X)~ ... ~Ap(X) ~ o.

In particular, the orbits of ~ = (xIAl(X» ... >Ap(X) > o} are of

the same type since we may infer that if X € ~ then (U,V) € GU(X)

if and only if

v {~ ~}

where W € O(n-p) and

follows that GU(X) = K

(±l}P x O(n-p).

U = diag(c 1 , ... ,c p )' ci € {±1}, i=l, ... ,p.

where K may be thought of as the group

Among the remaining orbits, we find one of particular interest,

namely

st(p,n) = (X € gl(p,n) IA 1 (X) = ... = Ap(X) 1}

* = (X € gl(p,n) Ixx = Ip} .

This is known as the stiefel manifold and plays a fundamental role in

orientation statistics, see e.g. Downs (1972) and Khatri and Mardia

(1977). Cf. also example 8.8. 0

Examl2le 2.8. Consider the set t of matrices of gl(p,n) p x n

It

rank p, p :: n. This is an open subset of gl(p,n). The group GL(p)

acts on gl(p,n)t by

GL(p) x gl(p,n) t ~ gl(p,n) t

(A,X) ~ AX .

The rows of X span a p-dimensional subspace of ~n. Any other set of

p vectors spanning the same subspace is given by the rows of AX for

some A € GL(p). It follows that the set of p-dimensional subspaces of

~n can be identified with GL(P)\gl(p,n)t. This is known as the

Grassman manifold and is denoted by G(p,n). The Grassman manifold and

10

similar objects plays a central role in the development of stereologi­

cal procedures, see Santa16 (1979). For instance it is of interest to

define a uniform distribution on G(p,n). It seems natural to seek for

a distribution which is invariant under rotations of the subspaces,

since this corresponds to choosing a subspace which is oriented at

random. This leads to considering the action of O(n) on G(p,n)

given by

O(n) x G(p,n) ~ G(p,n)

* (U, GL(p)X) ~ GL(p)XU

This action is transitive, which means - as we shall see in example 4.6

- that the uniform (invariant) distribution is uniquely determined. 0

A subset H of a topological group G is a (topological) subgroup

of G if H is a subgroup of the group G and a closed subset of the

topological space G. Notice that if H is a subgroup of a topologi­

cal group G and H is an open subset of G then H is a closed

subset of G.

Let G be a topological group and H a (topological) subgroup of

G. Then H acts on G in two ways.

The left action 0H of H on G is defined by

0H: H x G ~ G

(go,g) ~ gog·

The homeomorphism 0H(gO) is called left translation by go and is

often denoted by L go

It is clear that this action is free, and the

orbits are the right cosets Hg, g € G. If u:G ~ G is an orbit

representative with respect to this action, then letting U = u(G) we

have that

G HU

i.e. every

h € Hand

g € G has a unique representation as g = hu,

u € U. We call (2.1) a right factorization of

respect to H.

Similarly the right action c H of H on G is given by

(2.1)

where

G with

11

-1 -+ ggo

and the right translation by is also denoted by

The orbits under this action are the left cosets gH, 9 € G. If

v:G -+ G is an orbit representative of ~H and V = v(G) then

G VH (2.2)

is a left factorization of G with respect to H.

We denote the cOllections of right and left cosets respectively by

H\G and G/H, i.e.

H\G = {Hg,g € G}, G/H = {gH\g € G}.

Since H is closed, G acts transitively on both G/H and H\G

by

G x G/H -+ G/H (2.3)

(g,g'H) -+ gg'H

and

G x Ji'G -+ Ji'G (2.4)

(g,Hg' ) -+ -1 Hg'g •

We must require that H is closed to ensure the continuity of (2.3)

and (2.4). We shall refer to (2.3) and (2.4) as the natural actions of

G on G/H and Ji'G.

Similarly, in the case where G is a topological group acting

transitively on a topological space ~ and H is the isotropic group

of some Xo €~, we will be interested in a left factorization of G

with respect to H, because ~ is in one-to-one correspondence with

G/H by the mapping gH -+ gxo' Several examples illustrating the con-

struction of such factorizations are given in the next chapter.

We will now list a set of topological regularity conditions which

become relevant in connection with invariant measures.

12

A topological space ~ is said to be locally compact if

(i) for all x €~, there exists an open neighbourhood U of

x such that the closure of U is compact

(ii) for all pairs x,y €~, x # y, there exist open

neighbourhoods U of x and V of y such that U and

V are disjoint (Hausdorff condition).

We will call ~ an LCD-space if ~ is a locally compact topological

space with a denumerable base of open sets. Quite importantly, the

latter condition allows us to identify the concepts of Radon measures

and regular abstract measures on ~,

1978) •

(cf., for instance, Andersson,

Suppose that G acts on ~ and that G and

Below we present a condition which ensures that

Consider the mapping

f: G x ~ ~ ~ x ~

(g,x) ~ (gx,x)

and let

v = f(G x ~) = {(gx,x) Ig € G, x € ~}.

are LCD-spaces.

is an LCD-space.

(2.5)

(2.6)

We then define (G,~) to be a standard transformation group if

(i) G and ~ are LCD-spaces

(ii) v is a closed subset of ~ x ~

(iii) every compact subset of v is the image under f of a

compact subset of G x ~.

with this definition we have

Theorem 2.1. Suppose (G,~) is a standard transformation group.

Then

(i) G'~ is an LCD-space

13

(ii) the mapping

is a homeomorphism for all x € ~

(iii) the orbit Gx is closed in ~ for all x € ~. D

A subclass of the standard transformation groups is obtained by

requiring that the action of G on ~ is proper, which means that the

mapping r defined by (2.5) is proper. In general a mapping f from

one topological space to some other topological space is said to be

proper provided that f is continuous and the inverse image under f

of every compact set is compact. In case of a proper action it follows

that theorem 2.1 is true and in addition, it can be shown that

( iv) the isotropic group G = {g € G!gx x x} , is compact, for

all x € ~.

Another theorem, which is a useful corollary of theorem 2.1, is

given by

Theorem 2.2. Let G be a topological group and H a closed sub-

group of G.

G/H [ll'G]

If G is an LCD-space then the left [right] coset space

is an LCD-space. D

The following lemma concerns the question of existence of a measur­

able orbital decomposition.

Lemma 2.1. Let (G,~) be a standard transformation group.

Then there exists a mapping

(z, u): ~ ~ G x ~

so that x = z(x)u(x), where u(gx)

(z,u) is Borel-measurable.

u(x), g € G, and so that

D

In the case where ~ has constant orbit type, we may improve the

lemma as follows.

14

Lemma 2.2. Let (G,~) be a standard transformation group and sup­

pose that ~ has constant orbit type. Then there exists a Borel measur-

able orbital decomposition. 0

It is worth stressing, that all of the considerations carried out in

this section, also hold for the opposite group GO of G, which is

defined to be a topological copy of G endowed with the multiplication

rule

If G acts on ~ then GO acts on ~ by the prescription

-1 ~Go(g): x ~ ~G(g ) (x).

(2.7)

(2.8)

Let I:G ~ GO

then HO = I(H)

denote the identity map. If H is a subgroup of G,

is a subgroup of GO, and HO'Go is homeomorphic to

G/H, showing that we can restrict our discussion to left coset spaces.

Also, in connection with the theory of invariant measures, to be dis­

cussed in the next section, the concept of the opposite group turns out

to be a valuable tool.

Bibliographical notes

An extensive and general treatment of groups and actions is found in

Bourbaki (1970), chapter 1. Husain (1966) gives a most readable intro­

duction to topological groups, which also covers results on locally

compact groups. The proof of theorem 2.1 is given in Eriksen (1989) and

is based on results extracted from Bourbaki (1960), chapter 3. Lemma

2.1 is essentially a reformulation of Jespersen (1985), theorem 2.2.

The theorem by Jespersen is based on a major result of Effros (1965)

concerning the existence of a measurable orbit representative. Lemma

2.2 is obtained by a slight modification of the proof of lemma 2.1. The

details may be found in Eriksen (1989). The question of existence of a

measurable orbital decomposition has also been considered by wijsman

(1967, 1986) in some special cases, with the purpose of describing the

distribution of maximal invariant statistics.

15

3. Matrix Lie groups

This section contains a brief introduction to matrix Lie groups. The

focus is on developing tools for the construction of factorizations of

a group with respect to some subgroup. In this connection the Lie alge­

bra and the exponential map are the central concepts. It may be noted

that all the groups occurring in the examples considered in these notes

are matrix Lie groups.

Throughout the rest of these notes we often consider an m-dimension­

al differentiable manifold M and we then use the following notation.

A chart around p € M is a pair (U,~) consisting of an open neigh­

bourhood U around p and a diffeomorphism ~: U ~ Rm. Letting ~ -1 . denote the inverse mapping of ~, i.e. ~ = ~ and sett1ng V

~(U) S Rm we refer to the pair (V,~), or sometimes just to ~, as a

(local) parametrization of U.

A group G is said to be an m-dimensional Lie group if

(i) G is an m-dimensional differentiable manifold

(ii) the mapping

G x G ~ G

is smooth.

Condition (ii) is equivalent to smoothness of the mappings

and (gl,g2) ~ glg 2·

-1 g ~ g

Example 3.1. GL(k) - the group of invertible k x k matrices - is

a Lie group:

k 2 GL(k) is an open subset of m , and may thus be considered as a

k 2-dimensional differentiable manifold with the same differentiable

k 2 structure as m

Multiplication and inversion are infinitely often differentiable

mappings, i.e. they are smooth. D

A subset H of a Lie group G is called a Lie subgroup if

16

(i) H is a submanifold of G

(ii) H is a subgroup of G

(iii) H is a topological group.

Note that H need not be a topological subgroup. However, we have

the following theorem.

Theorem 3.1. Let G be a Lie group and H a closed subgroup of

G. Then H is uniquely structured as a topological Lie subgroup of

G. 0

Example 3.2. Invariance subgroups. Let ! € GL(k) and let

G (g € GL(k) Ig!g* = !}.

It is easy to verify that G is a group and, since the mapping

~: G -> G

* g -> g!g

is continuous, it follows that G = ~-1({!}) is closed, i.e. G is a

topological Lie subgroup of GL(k).

In particular, this applies to the case where k p+q and

I p,q

the group thus determined being the pseudo-orthogonal group of order

~ which is denoted by O(p,q}, i.e.

O(p,q) (A € GL(p+q) IAI A* = I }. p,q p,q

It is clear that if A € O(p,q) then det(A) € {±1}. This gives rise

to the definition of the special pseudo-orthogonal group of order

~ as

SO(p,q) (A € GL(p+q) IAI A* p,q Ip,q' det(A) 1}.

17

It can be shown that the connected component of SO(p,q) containing

the identity element is

sot (p,q) * (A € GL(p+q) IAI A p,q Ip,q' det(A) 1, det(All»O}

where All € GL(p) is determined by

A

In case q = 0 the groups O(p) = O(p,O), SO(p) = SO(p,O) are simply

the orthogonal group and the special orthogonal group, respectively. 0

A finite-dimensional vector space Lover m is called a (real)

Lie algebra if there exists a rule of composition (A,B) ~ [A,B] in L

satisfying

(i) [aA+f3B,C] = a[A,C] + f3[B,C] a,{3 € m

(ii) [A,B] = -[B,A]

(iii) [A, [B,C]] + [B, [C,A]] + [C, [A, B]] = O.

Condition (iii) is known as the Jacobi identity, and

called the Lie product of A and B. A subspace N of

subalgebra if A,B € N ~ [A,B] € N. For 0 # A € Land

has by (i) and (ii)

[aA,{3A] a{3[A,A] -a{3[A,A]

so that

[aA,{3A] o.

[A,B] is

L is a (Lie)

a,{3 € m one

This shows that everyone-dimensional subspace of L is a trivial

subalgebra.

Consider the vector space gl(k) of real k x k matrices and the

commutator product

gl(k) x gl(k) ~ gl(k)

(A,B) ~ AB - BA.

18

This is a Lie product making gl(k) into a Lie algebra.

The exponential map is the mapping defined by

exp:gl(k) ~ GL(k)

A ~ exp(A) ,

where

exp(A)

Let A,B € gl(k). Simple calculations show that

and

* exp(A) * exp(A )

exp(A+B) = exp(A)exp(B) if AB

In particular, for B = -A

exp(A)exp(-A) = exp(O) = I;

thus exp(A) belongs to GL(k) and

exp(A)-l = exp(-A).

BA.

For S € GL(k) and A € gl(k) it is seen that

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

Turning back to the groups, define a group to be connected, if it is

not the union of two non-empty and disjoint open sets.

The following theorem characterizes the Lie algebra of a Lie sub­

group of GL(k).

19

Theorem 3.2. Let H be an m-dimensional Lie subgroup of GL(k).

Then there exists an m-dimensional subalgebra h of gl(k) such

that the following holds.

Let A € gl(k), A "I- O. Then we have

HA (exp(As) Is € ffi} is a (connected) one-dimensional Lie

n subgroup of H

A € h.

Every connected one-dimensional Lie subgroup of H has the form HA

for some A € h, and h is called the Lie algebra of H. Conversely,

if h is a subalgebra of gl(k), then there exists a Lie subgroup of

GL(k) having h as its Lie algebra. 0

It is seen from (3.4) that HA is always a connected one-dimension­

al Lie subgroup of GL(k) and that for A "I- 0 we have HA = HAA •

The space gl(k) is, in fact, the Lie algebra of GL(k).

Remark. Different subgroups can have the same Lie algebra. This is

illustrated by taking a subgroup H which is not connected and letting

He be the connected component of H containing the identity. Then He

is a subgroup and the connected one-dimensional subgroups of Hand

He must coincide, i.e. the algebras coincide. This is exemplified by

GL(k), where the identity component is GL+(k) which consists of the

matrices with positive determinant. o

Example 3.3. The Lie algebra of O(p.g). Let I € GL(p+q) p,q be

the matrix

I - P {I oj

p,q 0 -I

and consider the pseudo-orthogonal group of order (p,q)

O(p,q) (B € GL(p+q) IBI B*=I }. p,q p,q

20

This is a Lie group and for A E gl(p+g) we get

* exp(As) I exp(As) n p,g

I exp(As)I-1 nP,g p,g

I p,g

*

S E IR

-1 exp(I AI s) p,g p,g exp(-A s) s E IR

so that the Lie algebra consists of the matrices o(p,g) =

(A E gl(p+g) II A=-A*I ). If g = ° then o(p) = o(p,O) is the p,g p,g

set of skew-symmetric matrices. 0

The Lie algebra of an m-dimensional subgroup H of GL(k) can

often be obtained easily in the way just illustrated. Alternatively,

let (U,~) be a chart containing I, i.e. U is an open subset of H

containing the identity element and ~ maps U diffeomorphically onto

an open subset V of IRm. Let ~ = ~-1, the inverse mapping of ~, and let u o = ~(I). Then the vectors a~:lv=v' i = 1, ... ,m, form a

~ ° basis of the Lie algebra, i.e. the Lie algebra is in fact the tangent

space of H at I.

Example 3.4. Let

t .. ~J

0, i > j}

denote the group of k x k upper triangular matrices with ones in the

diagonal.

Then the mapping

~: IR k (k-1)/2 ~ T (k) +1

(tij ) i<j ~ {tij }

is such that ~ ~-1 determines a chart with ~(O) I, and

21

Thus the Lie algebra t+1 (k) of T+1 (k) consists of the k x k upper

triangular matrices with zeroes in the diagonal. []

Next we discuss the question of factorization of Lie aroups. Let G

and H be closed subgroups of GL(k) with Lie algebras ~ and ~.

Suppose H is a subgroup of G implying that ~ is a subalgebra of

~. As mentioned in section 2 we are often interested in factorizations

of G with respect to H. In the following we illustrate a construc­

tion technique, which is applicable in most cases. Let ~ denote a

complement to ~ in ~, i.e. ~ can be written as the direct sum

~ ~ $ ~.

If K = exp(~) then we often have that G

hope that the mapping

~ x H -+ G

(a,h) -+ exp(a)h

HK KH. One could also

was a diffeomorphism, but in general the exponential map is not one-to­

one. A simple example is provided by

[[0 9]] [cos 9 exp -9 0 _ sin 9

sin 9]

cos 9 9 E IR.

However, in most cases we can find an open subset u of ~ so that

the mapping

u x H -+ G

(a,h) -+ exp(a)h

is differentiable, one-to-one and onto G, except maybe for a closed

null set (a submanifold of dimension less than the dimension of G).

The technique may be generalized as follows.

Let

~

and define

22

>/I: :fli ••• i:fq -+ G

(a l , ... ,aq ) -+ exp(a l ) ... exp(aq ).

We might then seek for open subsets u i C :f i , i

the mapping

uli ••• iuq x H -+ G

(a l , ..• ,aq,h) -+ >/I (aI' ... ,aq)h

is differentiable, one-to-one and (almost) onto G.

We now turn to some examples.

l, ... ,q, so that

Example 3.5. A factorization of SO(p). The special orthogonal

group SO(p) = (U € GL(p) Idet(U) = 1, uu* = I} acts linearly and p

transitively on the unit ball Sp-l = {x € IRP lllxll = I} by the law

SO(p) x Sp-l -+ Sp-l

(U,x) -+ Ux.

Let x~ = (0, ••• ,0,1) € IR P . The isotropic group K of Xo consists

of the matrices

{~ ~}, U € SO(p-l),

i.e. K is isomorphic to SO(p-l), and we will, with a slight abuse

of notation, write SO(p-l) for K. The Lie algebra of SO(p) is the

set so(p) of p x p skew-symmetric matrices. It is seen that

so(p)

where

{[ Let

sO(p-l) i :f

a * -a

e.] 1 ,

a i 1, ... ,p-l ,

23

and define

R: 8 -+ SO(p)

(9 1 ,···,9p _1 ) ~ eXp(9p_1Fp_1)···eXp(91F1)

where 8 = {9 € ~p-11-v/2 < 9 i < v/2, i = 1, ••• ,p-2,

(-v/2,v/2) U (v/2,3v/2)}. It is easy to see that

I i - 1 0 0 0

0 cos 9. 1

0 sin 9. 1

exp(9 iF i) 0 0 I . p-l-1 0

0 -sin 9. 1

0 cos 9 i

9 1 € p-

i = 1, •.. ,p-2. This means that eXp(9iFi) is a rotation in the plane

spanned by the i'th and the last coordinate vector.

It follows that

>#1 1 : 8 ~ sp-1

9 ~ R(9)Xo

is a diffeomorphism of 8 onto SP-1,{(Xi ) € ~Plxp = O}, and that

>#I~l(X) is the set of polar coordinates of x. Furthermore, the map­

ping

R: 8 x SO(p-1) ~ SO(p)

(9,U) ~ R(9)U

is differentiable, one-to-one and 'almost' onto SO(p). [)

Example 3.6. Factorizations of Sol~. The linear action of the

group SOl(l,q) on Rq+l has been considered in example 2.6. The unit

hyperboloid Hq was characterized as sol (l,q)/SO(q), i.e. we are

interested in a left factorization of sol (l,q) with respect to

SO(q). The Lie algebra of SOl(l,q) is given by (cf. example 3.3)

so(l,q) = {[~ U€SO(q)}.

It follows that

so(l,q) ='-1 ED so(q)

where

The exponential map

:f1 -+ sol (l,q)

a -+ exp(a)

24

can be shown to be a diffeomorphism onto the set of boosts, i.e. the 1 elements in so (l,q), which are of the form

B

where

2 x2 1 +--1+x1

x2 x3

1+X1

If B(q)

2 Xq+1

1 + 1+X1

denotes the set of boosts, then

it can be shown that sol (l,q) has both a left and a right factoriza­

tion as SOl(l,q) = B(q)SO(q) = SO(q)B(q).

In relativity theory the group 0(1,3) is known as the Lorentz

group and the elements of B(3) are also called pure Lorentz transfor­

mations. Our interest in sol (l,q) and the above factorizations de­

rives from their importance for the hyperboloid exponential model, see

Jensen (1981) and example 8.7.

25

An alternative factorization is due to the socalled Iwasawa decompo­

sition of so(l,q) which is given by

so(l,q) = ~2 ~ so(q)

where

is a Lie subalgebra of so(l,q). If A(~)

exp(z(O,t» then

{ c~ s~: o}

. ~~ ... '?~. ~ ..... . ° : I q _1

and

C(t)

exp(z(~,O» and C(t)

Let P(~,t) A(~)C(t) and P(q) (P(~,t) I (~) E IRq}. Then

SOl(l,q) = P(q)SO(q)

determines a left factorization and

[c~ P(~,t)uo = s~

+ 1/2e~lItIl2l + 1/2e~lItIl2

t

parametrizes Hq . The group P(q) acts transitively and freely on

Hq • 0

26

Example 3.7. A factorization of SOI~. The notation introduced

in example 3.5 and example 3.6 will be used without reference. We will

now make a generalization of these examples.

Consider the (p+q) x (p+q) matrix

I p,q {I: O} . ~ ~ .. ~ . ~~~ ..

This defines a scalar product

Y E IRp+q.

p ~ 2,q ~ 1 .

* on * by x*y x I y, p,q

Let sol (p,q) denote the connected component of the pseudo­

orthogonal group of order (p,q), i.e. the group O(p,q) of linear

transformations of IR p+q leaving the symmetric form x*x invariant.

Then sol(p,q) has the matrix representation (cf. example 3.2)

SOl(p,q) = (A E GL(p+q) Idet A 1, det All> 0, * AI A = I } p,q p,q

where All E GL(p) is determined by

A {All A2l

A12} A22

.

The action of sol (p,q) on IR p +q is defined by

sol (p,q) x IRp+q -+ IRp+q

(A, x) -+ Ax .

Now let * - p xo - (0, ••• , 0, 1) E IR Then the orbit

of U o is the generalized hyperboloid

and the isotropic group of Uo is isomorphic to sol (p-l,q). so we

wish to make a left factorization of sol (p,q) with respect to

sol (p-l,q).

with a chance of confusion we define, in accordance with the pre­

vious examples,

27

R(9) {.~~~~ .• ~ ..• ~ .• }, 9 E a o : I • q

and

P(/L,t) { I p _1 : 0 }

•. ~ ...• ~ P(/L,t)

If Q(9,/L,t) R(9)P(/L,t) and Q(p-1,q) (Q(9,/L,t) 19 E a,

then

SOT(p,q) = Q(p-1,q)SOT(p-1,q)

except for a closed null set. Defining ~(/L,t)

have that

Q(9,/L,t)UO

parametrizes except for a null set.

Bibliographical notes

o

The theory of Lie groups is treated in many mathematical textbooks

on differential geometry, see for instance Cohn (1957) and Helgason

(1978). In particular, semisimple Lie groups is a very well described

class of Lie groups, but in a statistical context it does not seem

natural to impose semisimplicity. Our considerations on factorization

are more or less selfmade, but it may be noted that the existence of an

Iwasawa decomposition follows from semisimplicity, see e.g. Helgason

(1978) or Barut and Raczka (1980).

28

4. Invariant. relatively invariant. and quasi-invariant measures

In this section we discuss existence and uniqueness of invariant,

relatively invariant and quasi-invariant measures on a space ~ with

an acting group G. In particular, the left and right invariant

measures on G itself are considered, and several basic formulas re­

lating these are derived. Various disintegration formulas are also

presented.

In the sequel it is assumed that (G,~) is a standard transforma­

tion group, as defined on p. 12.

Let ~(~) denote the real-valued continuous functions on ~ with

compact support. A Radon measure on ~ is a linear functional ~:~(~)

~ R with the property that ~(f) ~ 0 for f > O. The set of such

measures will be denoted by ~(~). The measures on ~, which are

traditionally used in statistic, are regular (abstract) measures, i.e.

mappings n defined on ~, the a-ring generated by the compact sets

in ~, satisfying (i) n(B) ~ 0, (ii) n(A U B) = n(A) + n(B) for

A n B = 0, (iii) lim n(A ) n~ n

00

n(UA ) 1 n

for A1 S ... S An C •.• and (iv)

n(B) = sup{n(K) IK S B, K compact}. Under the present topological

conditions there exists a one-to-one correspondance between Radon

measures and regular measures given by

~(f) J f(x)dn(x) f € ~(~), ~

where the right-hand is an ordinary integral with respect to the ab­

stract measure n (cf., for instance, Andersson, 1978). As in the

classical abstract measure theory, a Radon measure ~ can be extended

to a larger class of functions called the ~-integrable functions. If B

€ ~ then the indicator function 1B is ~-integrable and the regular

abstract measure n is simply determined by

n(B)

since the two kinds of measures coincide under the topological regular­

ity conditions we adopt we do not distinguish between them in the sub­

sequent discussions and we use the common notation

It(f) f f(x)djJ.(x) , f € ~(It) , !l

29

where ~(It) is the vector space of It-integrable functions.

If g is a proper one-to-one transformation of !l onto !l we let

9JL denote the measure It lifted by g, i.e.

9JL(f) = It(f 0 g)

where 0 signifies composition of mappings. Further, suppose It is

absolutely continuous with respect to a measure v and let h denote

the corresponding density (or Radon-Nikodym derivative), i.e.

djJ.(x) = h(x)dv(x).

Then 9JL is absolutely continuous with respect to gv and we have the

important formula for transformation of the density

d(9JL) (x) -1 h (g x) d (gv) (x) •

A measure It on !l is said to be invariant relative to the group

G acting on !l if

9JL = It, g € G.

Here, for short, we write 9JL for ~(g)lt, the measure It lifted by

~(g). For the construction of invariant measures, as discussed in the

next section, it is convenient to introduce the more general concepts

of relatively invariant measures and quasi-invariant measures.

* Let ~ be a continuous mapping from G into the positive reals m+

such that

, , , ~(gg) ~(g)~(g), g,g€G,

i.e. ~ is a group homomorphism. A mapping of this kind is called a

multiplier on G, and a measure It on !l is said to be relatively

invariant with multiplier ~ if

g-llt = ~(g)lt, 9 € G, (4.1)

or, equivalently, in terms of differentials

30

-1 d(g JL)(x) = l«g)dJL(x).

We shall often denote such a measure by JLl(. Note that an invariant

measure is a relatively invariant measure with multiplier l( = 1.

On the other hand, if JL ~ 0 is a measure fulfilling (4.1) for some

function l(, then l( is a multiplier in relation to which JL is

relatively invariant.

For relatively invariant measures one has the following existence

and uniqueness theorem

Theorem 4.1. Suppose G acts transitively and properly on ~.

Then for every multiplier l( on G there exists one and, up to multi­

plication by a positive constant, only one relatively invariant measure

JLl( on ~ with multiplier l(. 0

Let l( be a multiplier and let m be a positive continuous func­

tion on ~ which satisfies

m(gx) l«g)m(x), g € G, x € ~.

Then m is called a modulator with (associated) multiplier x. When

it is important to make the dependence of m on l( explicit we write

for m.

concerning the existence of modulators, we have the following the­

orems.

Theorem 4.2. If G acts properly on ~, then to every multiplier

l( there exists a modulator m having l( as its associated multi-

plier.

A subgroup K of

implies that gKg-1

type G/K with K

o

G is said to be regular, if 9K9-1 ~ K, g € G,

K. Furthermore, if (G,~) has constant orbit

regular we say that (G,~) has regular orbit type.

Theorem 4.3. Suppose that (G,~) has regular orbit type, where the

orbits are homeomorphic to G/K.

Let l( be any multiplier fulfilling

l( (k) 1, k € K.

31

Then there exists a modulator with ~ as its associated multi-

plier. o

The importance of the concept of modulator lies in the fact that by

means of modulators it is possible to construct any relatively invari­

ant measure - in particular, an invariant measure - on ~ from any l(0

other relatively invariant measure on ~. Specifically, suppose ~

is relatively invariant with multiplier ~o and that we wish to find a

measure ~l( which is relatively invariant with some other multiplier

l(. Let m -1 be a modulator with associated multiplier l(l(~1 (where l(l(0

-1 -1 l( (l(l(o ) (g) = l«g)l(o(g) ). Then, as is simple to check, the measure ~

given by

l(o m -1~

l(l( 0

is, in fact, relatively invariant with multiplier ~.

(4.2)

Example 4.1. GA+C1l-invariant measure. Let the setup be as in ex­

ample 2.3 which is concerned with the location-scale group. Let A be

the restriction to ~ = (x € ~nls(x) > O} of Lebesgue measure on ~n. Evidently, A is relatively invariant with multiplier

Furthermore, it is easily verified that m(x) = s(X)n

with l( as associated multiplier. Hence

~(x) = s(x)-n dA(x)

is an invariant measure on ~.

l«(a,~» = n a .

is a modulator

o

Example 4.2. CCn,Hl-invariant measure. In the setup of example 2.4,

let A be the restriction to ~ = (x € ~nls(x) is positive definite}

of Lebesgue measure on ~n. It is clear that A is relatively invari­

ant with multiplier l«A) = IAI. Furthermore, m(x) = Is(x) 11/2 is

easily shown to be a modulator with l( as associated multiplier. It

follows that

~(x) = Is(x) 1-1/ 2 dA(x)

is an invariant measure on ~. o

32

By the above mentioned existence and uniqueness result for invariant

measures there exist measures a and ~ on G which are invariant,

respectively, under left action 0 and right action c of G on

itself, and these measures are termed left invariant and right invari­

ant, respectively. (Alternatively, a and ~ are called left and

right Haar measure). Below we show that there exists a multiplier A,

called the modular function or the module of G, such that

a = A~ (4.3)

(with suitable choice of the arbitrary multiplicative constants for a

and ~). When we wish to make the dependence of a, ~ or A on G

explicit we use the notations a G, ~G and AG• It follows that a is

relatively invariant under c with multiplier A-I and that ~ is

relatively invariant under 0 with multiplier A-I. We also note, and

later prove, the important formula

ff(g-l)da = ff(g)d~,

and the relations

and

-1 J( ~

-1 ~(J(A)

(4.4)

(4.5)

(4.6)

(4.7)

where for instance (4.7) means that a relatively right invariant

measure with multiplier (J(A)-l is relatively left invariant with

mul tiplier J( .

Proofs of formulas (4.3)-(4.7). Let o (g) and -1

c(g ) denote left

and right translations by g, respectively. Obviously, 0 0 c = coo,

Le.

To prove (4.3), first note that c(g)a is left invariant, as appears

from the calculations

o (g') (I'. (g)a) (f)

33

(I'.(g)a)(f 0 o(g'))

a(f 0 o(g') o I'. (g))

a(f 0 I'. (g) oo(g'))

(0 (g' )a) (f 0 I'. (g))

a(f ol'.(g))

I'.(g)a(f).

Applying theorem 4.1 and the remark preceding it we find that there

exists a multiplier A such that

I'.(g)a = A(g)a,

-1 i.e. a is relatively right invariant with multiplier X = A .

Letting Xo = 1, formula (4.3) follows from (4.2) on noticing that

m(g) = A(g) is a modulator corresponding to the multiplier A-1 = -1

XXO; in fact

m(l'.(g)g') -1 -1 m(g'g ) =A(g) m(g').

To prove (4.5) we observe that

--1 - --1 d(o (g ) (xa)) (g) = X (gg)d(o (g )a) (g)

X (g) X (g) da (g)

= X (g)d(xa) (g).

Formula (4.6), which is proved similarly, implies that

-1 ~(xA) = XA~ = xa

and this proves (4.7). Finally, for f € ~(G)

v -1 f(g) = f(g )

v v and let a(f) = a(f). using

v V (fol'.(g)) =foo(g)

it follows that

v let f be defined by

34

v v v v v V E.(g)a(f) a (f 0 E. (g) ) a(f o c5 (g» = c5(g)a(f) a (f) a (f)

v showing that a is right invariant.

v Consequently a (f) (3 (f) , which

is equivalent to (4.4). 0

The group G is said to be unimodular if A (g) = 1, g E G. If a

group is compact or commutative it is unimodular. Subgroups of unimodu­

lar groups are not in general unimodular (cf. example 4.3 below). A

general method for calculating modular functions will be given in sec­

tion 6, see formula (6.8).

Example 4.3. Triangular group. The triangular group T+(n) is the

group of n x n upper triangular matrices with positive diagonal ele­

ments. It is a subgroup of the general linear group GL(n) , and the

latter is unimodular as shown in example 6.1. However, the module of

T+(n) is, as will be proved in example 6.2,

n 2i-n-1 II t ..

i=l 11

where tii denotes the i-th diagonal element of T E T+(n). Hence

T+(n) is not unimodular. o

More generally, suppose that g-l~ and ~ are equivalent, (i.e.

mutually absolutely continuous) for every g in G. Then there exists

a nonnegative function X on G x ~ such that

-1 g ~ = X(g,·)~

or, written in terms of differentials,

-1 d(g ~)(x) =X(g,x)<4t(x).

Transforming this identity by g we obtain

, , X (gg, x) <4t (x) X (g,gx)X (g,x)<4t (x).

This leads to defining a quasi-multiplier to be a positive continu­

ous function

35

with the property that

I I

x(gg,x) = x(g,gx)x(g,x)

and Jl is said to be a quasi-invariant measure with quasi-multiplier

X if

-1 d(g Jl) (x) x(g,x)dJ.L(x), 9 € G, x € ~.

Theorem 4.4. Suppose G acts transitively on ~, let u be an

element of ~ and let K denote the isotropic group at u, Le. K

G . u Then K is closed, ~ and G/K are in homeomorphic correspon-

dence, and

(i) There exists a quasi-invariant measure Jl on ~ with some

quasi-multiplier x. Any two quasi-invariant measures are equivalent,

and any quasi-multiplier determines the corresponding quasi-invariant

measure uniquely, up to a multiplicative constant.

Furthermore, letting ~ denote the mapping from ~ to G/K = {gK:g € G} which establishes the homeomorphic connection between these

two spaces, we have

(ii) A positive continuous function X on G x ~ is a quasi-multi­

plier for some quasi-invariant measure Jl on ~ if and only if X is

of the form

x(g,x) p(gz)/p(z), z€~(x), (4.8)

where p is a positive locally integrable function on G satisfying

p (gk) AK(k) x-(k) peg), k € K, 9 € G.

G (4.9)

(Note that, because of (4.9), the right hand side of (4.8) is the same

whichever z in ~(x) is chosen.)

(iii) A quasi-invariant measure Jl and the associated p-function

(as described in (ii) above) are related by

J f(g)p(g)daG(g) G

where v = 'J.L.

36

J J f(gk)daK(k)dv(gK) G/K K

(4.10 )

o

It is to be noted that if G is a group and if K is an arbitrary

closed subgroup of G then the above theorem applies with ~ = G/K

and ~ as the natural action of G on G/K.

Corollary 4.1. Suppose G acts transitively on ~ and let K be

an isotropic group of this action. Then there exists a relatively in­

variant measure J.L on ~ with multiplier X if and only if

X (k) (4.11)

This measure is unique up to a multiplicative constant and v = 'J.L

(defined in theorem 4.4) satisfies

J f(g)x(g)daG(g) G

J J f(gk)daK(k)dv(gK). G/K K

o

If the action of G on ~ is proper then K is compact (cf. (iv)

of the remark after theorem 2.1) and hence AG(k) = AK(k) = 1, k € K,

* since the only compact subgroup of (ffi+,.) is {1}. It follows, in

this case, that to every multiplier X on G there exists a unique

(except for a constant) relatively invariant measure on ~ with X as

the associated multiplier.

Corollary 4.2. Supppose G acts transitively on ~ and let K be

an isotropic group of this action. Then there exists an invariant

measure J.L on ~ if and only if

(4.12 )

This measure is unique up to a multiplicative constant and v = 'J.L

satisfies

J J f(gk)daK(k)dv(gK). G/K K

37

The same relation for the opposite group GO is seen to be equivalent to

f f f(kg)d~K(k)dvO(Kg). l<'GK

D

Proof. The first decomposition is a simple consequence of (4.10),

since we can choose p(g) = 1. NOW, considering the opposite group and

applying this decomposition we obtain

f f(g)da o(g) GO G

where denotes multiplication in GO. If I:Go ~ G is the identity

mapping, then I(a ) = ~G and I(a ) = ~K' Since we can identify GO KO

GO/KO and l<'G, the last assertion easily follows. D

Example 4.4. A generalization of Fubini's theorem. A subgroup K

of G is said to be normal if gkg-1 € K, k € K, 9 € G. If K is

normal then G/K = (gK:g € K) can be endowed with a group structure

by the following prescription of the group operation

, , (gK) (gK) ggK.

with this structure G/K is called the quotient group of G and K.

If G is a Lie group and if K is a closed normal subgroup of G

then G/K is a Lie group. Left invariant measure a G/ K on G/K is

clearly also invariant under the action of G on G/K. It follows

from corollary 4.2 that AG(k) = AK(k), k € K, and that

f f f(gk)daK(k)daG/K(gK). G/K K

In case G = V is a vector space and K L is a subspace, we may

identify V/L with a complement M to L in V, i.e. V = L ~ M.

We then recognize the well-known factorization, due to Fubini,

where ~ denotes Lebesgue measure and ® indicates product measure.

D

38

Example 4.5. Factorizations of GL(n). and existence of associated

invariant measures. The group GL(n) may be written as a product in

the following two fashions

GL(n) = O(n)T+(n) = T+(n)O(n).

Both O(n) and T+(n) are closed subgroups of GL(n), and GL(n)

and O(n) are unimodular while T+(n) is not (cf. examples 6.1, 6.5

and 4.3). Therefore, by corollary 4.2 there exists a GL(n) invariant

measure on GL(n)/O(n) = T+(n) but not on GL(n)/T+(n) = O(n).

If we consider the action of GL(n) on the space PD(n) of posi­

tive definite matrices defined by

GL(n) x PD(n) ~ PD(n)

* (M,~) ~ ~M (4.13)

then O(n) is the isotropy group of I E PD(n) and since GL(n) and

O(n) are both unimodular it follows from corollary 4.2 that there

exists a measure v on PD(n) which is invariant under the action

(4.13) . o

Suppose that G acts transitively on ~ and that the isotropic

group K of Xo E ~ is compact. As noted after corollary 4.1 we then

have AG(k) = AK(k) = 1, k E K. Applying corollary 4.2 we obtain that

~ has a G-invariant measure ~.

The compactness of K also implies that if f E ~(~) then

f (g ~ f(gxo» E ~(G)

and since the measure a on ~ given by

is invariant, it follows from the uniqueness that (except maybe for a

constant) a =~, i.e. the invariant measure ~ on ~ is given by

39

Example 4.6. Invariant measures on Step,nl and Gep,nl: existence

and uniqueness. The above applies, in particular, to the stiefel

manifold (example 2.7) and the Grassman manifold (example 2.8). In both

cases G is also compact, so that a G may be chosen as a probability

measure, and thereby ~ becomes the unique invariant probability

measure on !I • D

Example 4.7. sol e1,gl-invariant measure on Hq and Begl: exist­

ence. As discussed in example 3.6, sol (l,q) is both right and left

factorizable, as

sol (l,q) = SO(q)B(q) = 8(q)SO(q).

Consider the action of sol (l,q) on the unit hyperboloid

Hq {x (x x X)* € Rq+1 ·.x*x = 1, = = l' 2'···' q+ 1

given in matrix representation by

SOl(l,q) x Hq ~ Hq

(A,x) ~ Ax.

The isotropy group at (1,0, ... ,0) € Hq is SO(q) which is compact.

Furthermore, SOl(l,q) is unimodular (as will be shown in example

6.5). Hence (4.12) is satisfied and there exists a sol (l,q)-invariant

measure on Hq or, equivalently, on the set of boosts B(q). D

Example 4.8. SO(p,q)-invariant measure on HP,q: existence. Con-

sider (cf. example 3.7) the action of SO(p,q) on the generalized

hyperboloid

where the linear and transitive action is defined by

SO(p,q) x HP,q ~ HP,q

(A, x) ~ Ax. (4.14 )

40

The isotropic group at Xo € HP,q is isomorphic to SO(p-l,q), which

is noncompact if p > 1. However, SO(p-l,q) is a closed subgroup of

SO(p,q), and SO(p,q) is unimodular for arbitrary p and q (cf.

example 6.5). It follows that there exists an invariant measure on

HP,q under the action (4.14). D

Example 4.9. GA+Cl)-invariant measures. The location-scale group

GA+(l) may be identified with the group G of 2 x 2 matrices of the

form

g = [~ {] , a > 0, f € ~.

It is easily seen that the left and right invariant measures on G are

given by

-2 a dadf

and

-1 a dadf.

Consequently, by (4.3),

-1 a

showing that G is not unimodular.

Let K be the subgroup of G consisting of elements of the form

Here K is closed, noncompact, and unimodular (because it is commuta­

tive). Furthermore, AG(k) = 1 for every k € K, and consequently

there exists a G-invariant measure on G/K.

* Letting x E ~ be represented as the two-dimensional vector (x,l)

the action of GA+(l) on ~ with Xo = 1, as defined in example 2.3,

Le.

41

([a,f),x) ~ ax+f ,

becomes

and it is seen that K and G/K may be identified with the groups of

location and scale transformations of ffi, respectively.

Let K denote the isotropic group G(0,1)* of this action. The

elements of K are of the form

k [~ ~] a > o.

Since K is noncompact and unimodular (because it is commutative) we

may conclude, by the remark after theorem 2.1, that the action is not

proper and, by corollary 4.2, that there exists no invariant measure

under the action of G on ffi. 0

Bibliographical notes

The main theorem of this section is theorem 4.4, which is a repro­

duction of theorems 4.1.1 and 4.3.1 of Barut and Raczka (1980), which

however contains no proofs. It may also be extracted from Reiter

(1968), chapter 8, which gives a nice treatment of the concepts and

provides the necessary proofs. Theorem 4.2 is a consequence of proposi­

tion 2.4.7 of Bourbaki (1963), chapter 7 which also contains a general

treatment of the contents of this section. Theorem 4.3 is proved in

Eriksen (1989).

5. Decomposition and factorization of measures

Suppose a space ~ is partitioned into disjoint subsets ~v,

v € U,

measure

and let

on

f f (x) C4.t (x) ~

be a measure on &. If for each v € U we have a

and if there is a measure K on U such that

f f f(x)dp (x)dK(v) U ~ V

(5.1)

V

42

for every integrable function f then «pV)V€IT,K) is said to consti­

tute a decomposition of ~, and we speak of (5.1) as a disintegration formula. We have already in section 4 encountered decompositions and disintegrations, cf. formula (4.10) to which we return below.

Two types of questions arise here

(a) Given measures ~ and K on ~ and IT, respectively,

does there exist a family of measures (pv)v€IT such that

(5.1) holds.

(b) Given measures ~ on ~ and, for every v € IT, on

~v, does there exist a measure K on IT such that (5.1)

holds.

The answer to (a) is affirmative under very weak conditions (see for instance theorem 5.1 below and theorem 15.3.3 in Kallenberg (1983»,

and the real problem lies in constructing or describing the measures pv. On the other hand, a positive answer to (b) is available under

considerable restrictions only. In this case there is, in addition, the problem of constructing or describing K.

Next, let u and s be mappings defined on ~ and with range spaces ~ and ~, respectively, and let again ~ denote a measure on ~. One may then ask whether there is an associated factorization of the lifted measure (u,s)~ as

(u,s)~ K 8 p, (5.2)

where K and P are measures on ~ and ~, and one may seek to describe or construct K or p, or both.

This factorization problem is closely related to the decomposition

problem. Specifically, suppose u is a mapping of ~ onto IT such that ~v = {x: u(x) = v} for every v € IT. If, moreover, the level

sets ~v are all (or almost all) of the same mathematical character so

that each can be identified with a certain set ~,

measure on ~ and Pv the corresponding measure on

and (5.2) are essentially the same.

and if P is a

~v then (5.1)

The questions outlined above constitute in a sense the main techni-

cal problem of mathematical statistics. We shall be concerned here exclusively with exact and explicit solu-

43

tions to these questions, when the partition of ~ is generated by a

group G acting on ~, and we shall pay attention to the possibility

of characterizing K, or p, or the p~'s by invariance. Important-

ly, however, even when exact solutions are not available it is often

possible to obtain highly accurate approximate solutions, see Barn­

dorff-Nielsen (1988).

If ~ is a topological space, then ~(~) denotes the collection of

measures on ~, and for ~ € ~(~)- we let ~(~) denote the vector

space of ~-integrable functions on ~.

Suppose in the following that (G,~) is a standard transformation

group. A solution of question (a) is then available as

Theorem 5.1. Suppose that ~ € ~(~) is a-finite. Let K € ~(G\~)

be a a-finite measure, which is equivalent to ~~, ~ denoting the

orbit projection. Such a measure K exists, even though ~~ is not in

general a-finite. Then there exists a collection of measures (p~)~€G\~

such that

i) p~(x) € ~(Gx) K-almost every ~ € G\~

ii) ~ ~ p~(f) € ~(K) for every f € ~(~)

iii) J f(x)~(x) ~

whenever f €

J J f (y) dp ( ) (y) dK (~(x) ) G\~ Gx ~ x

~ (~) . o

Combining this theorem with theorem 4.3 we obtain, in generalisation

of corollary 4.2,

Corollary 5.1. The measure ~ is quasi-invariant with quasi-multi­

plier ~(g,x) if and only if for K-almost every ~(x) € G\~ the

measure p~(x) is quasi-invariant on Gx with quasi-multiplier

~(g,y), (g,y) € G x Gx.

Suppose that ~ is quasi-invariant with quasi-multiplier ~. Then

there exists an invariant measure on &, which is equivalent to ~,

if and only if

~(g,x) 1, g € G x

for ~-almost every x € ~.

(5.3)

44

Finally, if Gv is the isotropic group of some y E Gx, with v =

vex), then (5.3) holds if and only if

(5.4)

for K-almost every v E G\~. o

Proof. Suppose that ~ is quasi-invariant with quasi-multiplier

~(g,x). It is easy to see that this is equivalent to the statement

that for K-almost every vex) E G\~ we have that pv(X) is quasi-in-

variant on Gx with quasi-multiplier

A is invariant and equivalent to ~.

have

~(g,y), (g,y)E G x Gx. dA Let ~(x) = hex) > o.

hex) ~ (x) h(gx)~(g,x)~(x)

so that

hex) = h(gx)~(g,x).

It follows that (5.3) must be satisfied.

On the other hand, the quasi-multiplier ~(g,y) of

fies, according to (4.8) and (4.9),

~(k,y)

AG (k)

Y k • , E Gy . "G (k)

Suppose

Then we

sat is-

This shows the equivalence of (5.3) and (5.4). Suppose that (5.3) is

satisfied, and let (z,u) be a decomposition of the type described in

lemma 2.1. If m(x) = ~(z(x),u(x», then a bit of calculation shows

that m(gx) = ~(g,x)m(x) and it is now easy to see that m(x)-1~(x) is invariant. o

The function m is called a modulator with quasi-multiplier x. Note, that the last lines in the proof of corollary 5.1 contains a

method, under the assumption (5.3), for the construction of a modulator

on the basis of an orbital decomposition and ultimately for the con­

struction of an invariant measure. This method will be considered in

more detail in section 6.

45

A more explicit solution to (a) is available in terms of geometric

measures when ~ is a d-dimensional Riemannian manifold with metric

~. Recall that if Eix ' i = l, .•. ,d, denotes the coordinate frame at

x € ~ corresponding to a local parametrization (v,~) then the geomet­

ric measure 0 is given by

do (x) (5.5)

Here q is the d x d matrix with elements qij(V) = ~x(Eix,Ejx)

where x = ~(v) and ~ denotes Lebesgue measure on V.

In the special case where ~ is a submanifold of mk (endowed with

the inherited metric) the modulating factor in (5.5) may be calculated

as

(5.6)

Here denotes the (generalized) Jacobian 1 £» *£» 11/2, * where £»

is the d x k matrix * a-/J lav. Supppose that the sets are submanifolds of ~ determined as

the level sets of a differentiable mapping u from ~ to some other

Riemannian manifold ~. Letting ~ and

on ~ and ~, respectively, and writing

K

~u

be the geometric measures

instead of ~ we have .".

that the decomposition of ~ is determined by

1 *1-1/2 u pu = DuDu 0 •

Here u o denotes geometric measure on

(5.7)

~u and Du is the differen-

tial of the mapping u. In principle the determinant in (5.7) should

be calculated by representing Du as a matrix corresponding to (arbi­

trary) local orthonormal (relative to the Riemannian metric) bases for

~ and ~; see, however, exercise 19.

Next, we present a situation where a solution of (b) is feasible.

Suppose G acts properly on ~. Denoting the level sets of the

orbit projection .". by ~."., .". € G\~, we may for each .". € G\~ de-

fine a measure p.". € ~(~.".) by the prescription

46

f f(gx)d~(g), x E ~~, G

where ~ is the right invariant measure on

hand side is, in fact, the same whichever

(5.8)

G and where the right

x E ~ is considered. v

Theorem 5.2. Suppose G acts properly on ~, define measures p~

by (5.8) and let ~ be a measure in ~(~) such that

A(g)~, g E G, (5.9)

i.e. ~ is relatively invariant with multiplier A-I. Then there

exists a measure K on G'~ such that ((p~)~ E G,~,K) is a decompo-

sition of ~. The corresponding disintegration formula may be written

f f(x)~ = f f f(gx)d~(g)dK (5.10) ~ G'~ G

where ~ is right invariant measure on G. D

The measure K in theorem 5.2 is called the quotient measure and is

often denoted by ~/~. It should be noted that the measures ~ satis­

fying (5.9) are the only ones for which there exists a K such that

(5.10) is satisfied.

Note that in the particular case where the action of G on ~ is

free, so that ~ = G x ~/G, theorem 5.2 yields, in effect, a factori­

zation

(5.11)

Next, we give a similar decomposition without the properness condi­

tion, but where we suppose that. ~ has regular orbit type G/K and

that

(5.12)

Consider the subset ~o of ~ consisting of the points with isotropic

group K, i.e.

~o R}. (5.13)

47

Then we might hope for a decomposition onto G/K x ~o. However, in

general ~o does not represent the orbit space. Suppose that u € !to

and gu € !to' g € G. Then K = G gu -1 -1 gGug = gKg . This gives rise

to considering the normalizer H of K defined by

I -1 H = {g € G gKg K} • (5.14)

It follows that K is a normal subgroup of H and that the group H/K

acts freely on !to by

~H/K: H/K x ~o ~ ~o

(hK,u) ~ hu.

(5.15)

This suggests that we seek a decomposition onto G/K x (H/K)'~O.

First we need some preparations in order to state the theorem. Since

K is a normal subgroup of H, it follows that H acts on K by the

law

f:HxK~K

(h,k) ~ hkh-1 (5.16)

It is easy to see that a K is relatively invariant. Denote the multi­

plier by Xo' i.e.

(5.17)

In particular, we have that

(5.18)

From (5.12) and (5.18) it follows that

-1 -1 X (hK) = 4G(h) X 0 (h) (5.19)

is a well-defined multiplier on H/K. According to theorem 4.1 there

exists a modulator n on ~o with X as the associated multiplier,

i.e.

48

n(hu) x(hK)n(u), u € ~O' hK € H/K. (5.20)

If we let a G/ K denote the measure on G/K which is invariant under

the action of G then for each v € G\~

~(~v) by the prescription

we may define a measure P € v

f n(U(x))f(gKu(x))daG/K(gK), G/K

x € ~ v

(5.21)

where u: ~ ~ ~o is an arbitrary orbit representative. The definition

of is independent of u. We now have the following theorem.

Theorem 5.3. Let (G,~) be a standard transformation group of

regular orbit type and suppose (5.12) is satisfied. Define pv by

(5.21) and suppose ~ is an invariant measure on ~. Then there

exists a measure K on G'~ such that «pv)v € G,~,K) is a decompo-

sition of ~. The corresponding disintegration formula may be written

f f(x)~(x) ~

f f n(U(X))f(9KU(X))daG/K(9K)dK(v(x)). G'~ G/K

(5.22)

o

A straight-forward application of theorem 5.3 shows the following

factorization theorem.

Theorem 5.4. Suppose (G,~) is a standard transformation group of

regular orbit type, and let G act transitively on the LCD-space ~.

Furthermore, let s: ~ ~ ~ be a continuous mapping which commutes with

the actions of G on ~ and ~, i.e. s(gx) = gs(x), and suppose

(s,v) is proper.

If ~ is an invariant measure on & then

(s,v)~ p ~ K (5.23)

where p is the unique (up to a mUltiplicative constant) invariant

measure on ~ and K is a certain measure on the orbit space G\~.

o

49

If G acts properly on ~ and on ~ and s: ~ ~ ~ is continuous

and commuting with the action of

5.4 are satisfied. In this case K.

G, then the conditions of theorem

equals the quotient measure m-1~/~ where ~ is the right invariant measure on G and m is a modulator

on ~ with A as the associated multiplier.

Example 5.1. The action of SO(p,a) on a cone. Consider the cone

1}

and define rex) = (x*x) 1/2, x €~. Let ~ be Lebesgue measure on

~, which is an open subset of mp+q • The group SO(p,q) acts

linearly on mp+q and ~ is an invariant subset under this action.

Hence SO(p,q) also acts on ~ and ~ is invariant. Let ~ = HP,q

and sex) r(x)-lx . Then

(s,r)~ = p 8 K.

determines p as an invariant measure on HP,q. In particular,

I f(s)dp (s) HP,q

CD

I f 1 (r)dK.(r) I f 2 (s)dp(s). o HP,q

I f(s(x»dx. {x€~lro~r(x)~r1}

1. Then

If, for instance, q = 0

the invariant measure on

this is the well-known characterization of Sp-1 given by

P (A) A ({x € mPlo < "x" p -1

~ 1, "x" X € A}). o

In some situations it is possible to characterize the measure K. in

theorems 5.1-5.4 by invariance as will now be discussed.

Suppose that (G,~) is a standard transformation group. As we shall

see in a moment, there are instances, where there exists a 'supplement­

ary' group H acting on ~ by (h,x) ~ xh-1 , say. Combining this

with the action of G, we obtain an action of G x H on ~ given by

50

(G x H) x 3: -+ 3: -1

«g,h),x) -+ gxh . (5.24)

By referring to the preceeding theorems, it is easy to establish the

following theorem.

Theorem 5.5. Suppose that the action (5.24) is transitive and that

G (respectively H) acts transitively on the LCD-space ~ (respec­

tively ~) and in addition that

(s,r): 3: -+ ~ x ~

is a proper mapping with the property that

-1 s(gxh )

-1 r(gxh )

gs (x)

hr(x) .

Furthermore, suppose that both of the actions of G and of H on 3:

- in turn - fulfills the conditions of either theorem 5.2 or theorem

5.4.

Then if ~ is invariant on 3: under the action of G x H we have

that

(s,r)(~) = a ® p

where a (respectively p) is invariant measure on ~ (respectively

~) . o

Remark. Observe that both a and p may be interpreted as quo­

tient measures.

Example 5.2.

-1 (T, x) -+ T X

Consider example 5.1. The group * IR+ acts on 3:

Furthermore, r(x)-(p+q) ~(x) is invariant under the action of

* SO(p,q) x IR+, since ~ is relatively invariant with multiplier

by

that we may consider the action of

-1 (T,r) -+ T r.

51

The last equality also implies

* on ~ = r(~) = ffi+ given by

Applying theorem 5.5 we obtain that

(s,r) (r(x)-(p+q)djL(X)) = da(s)dp(r)

where a

spectively

(respectively

* ffi+) .

p) is the invariant measure on (re-

o

Example 5.3. Consider examples 2.3 and 4.1 and the action of G

GA+(1) = ffi: x ffi on ~ = (x € ffinls(x) > O} given by

and

ant

[a,f]: X -+ ax + fxO

<','> an inner product on ffin.

u: ~ -+ ~ = {x€ffinlx=o, s(x)=1}

u (x) -1 -s (x) (x-xxo )

Furthermore, the maximal invari­

is g'iven by

Let x € ffin be considered as a row vector and define

where

O(n) (V € GL(n) I <xV,yV>

is the group of isometries.

The measure djL(x) = s(X)-n dA(x) is invariant under the action of

G x H and it follows from theorem 5.5 that

52

«s,X) ,u) (IL) a ® p

where a is left invariant on G and p is invariant on ~ under

the action of H. In fact, a bit of reflection reveals that ~ ~

O(n-1)/O(n-2), Le. can be considered as a sphere in n-1 IR , which

is also clear by noting that ~ is the intersection between the sphere

{x € IRnl<x,x> = 1} and the hyperplane {x € IRnl<x,xo> = a}. 0

Another situation where the quotient measure can be characterized by

invariance, is obtained by considering a closed subgroup H of a group

G and supposing that

Then, for every g € G and h € H,

which shows that AG is a modulator on G with respect to left action

°H of H on G. Hence, by (4.3) , the measure -1 AG a G f3 G is a rela-

tively invariant with multiplier -1 and we may there-measure on G AH

fore use theorem 5.2 to decompose this measure. In fact, (5.11) applies

so that f3 G is factorized as f3 G = f3 H ® K with K being the quotient

measure f3 G/f3 H • Comparing this to corollary 4.2 we see that f3 G/f3 H

must be invariant measure on H\G, relative to the natural action of

G on H'G. There is, of course, a similar factorization of a G, and

we may summarize these comments in the two formulas

G = H x G/H

a G = a H ® aG/aH (5.25)

and

G = H x H\G

f3 G = f3 H ® f3 G/f3 H (5.26)

53

where aG/aH and ~G/~H are measures on G/H and H\G, invariant

under the natural actions of G on G/H and H\G, respectively.

Bibliographical notes

A general treatment of disintegration of measures is given in Bour­

baki (1959). Theorems 1, 3, 4 and corollary 1 of this section are

proved in Eriksen (1989). Except for the conditions of orbit regularity

theorem 4 generalizes lemma 3 of Andersson, Br0ns and Jensen (1983),

cf. also Barndorff-Nielsen, Bl~sild, Jensen and J0rgensen (1982).

Example 5.1 shows a situation, which is not covered by this lemma, but

where the conditions for applying theorem 5.4 are fulfilled. Theorem

5.2 is a reproduction of proposition 2.2.4 of Bourbaki (1963), chapter

7. Finally, this section also contains a few remarks on Riemannian

geometry and we refer to Boothby (1975) for an excellent introduction

to Riemannian geometry and to Tjur (1980) for more details concerning

the decomposition of geometric measures.

6. Construction of invariant measures

We shall discuss here methods of constructing a G-invariant measure

~ on the space ~, in the sense of expressing ~ on the form

~(x) = ,(x)dX(x) (6.1)

for some function , and where X is some given measure on ~. If ~

is an open subset of Rr then X will typically be Lebesgue measure.

More generally, X may be the geometric measure on ~ when ~ is a

Riemannian manifold (formula 5.5)).

The construction of , can often be achieved as follows. One starts

by showing that X is quasi-invariant under G with quasi-multiplier

x(g,x), say. supposing that the conditions of corollary 5.1 are ful­

filled, we know that there exists an invariant measure of the form

(6.1). Essentially we have to check that

X(k,x) 1, x €~, k € Gx .

Subsequently we seek a modulator m with quasi-multiplier x, i.e. a

positive function m on ~ for which

54

m(gx) l( (g,x)m(x).

Finally, we define .(x) as l/m(x) and then ~, given by (6.1), is

invariant. The problem is thus reduced to constructing a suitable

modulator m. The construction method on which we will concentrate,

has, in fact, already been applied at the end of the proof of corollary

5.1, i.e. if (z,u): ~ ~ G x ~ is an orbital decomposition as in lemma

2.1, then m(x) = l«z(x),u(x» is a modulator.

More specifically, we will treat the case where ~ is a subset of

mn and where G acts linearly on ~, i.e. {~(g) Ig € G} are linear

transformations of mn leaving ~ invariant. Moreover, we will assume

that the following regularity conditions are satisfied

i) ~ is an m-dimensional differentiable submanifold of mn

ii) G = {~(g) Ig € G} is a closed subgroup of GL(n).

The last condition implies that G is itself a differentiable submani­

fold of the vector space of n x n matrices. For simplicity we will

assume that ~ is covered by a single chart, i.e. there exists an open

subset V of mm and ~:V ~ mn so that ~ is a parametrization of

~ and ~ has differential ~ of rank m. The geometric measure A

on ~ is given by

dA (x)

where Am is Lebesgue measure on V and J~(V) = I~*(v)~(v) 11/2,

cf. (5.6). The mapping f(g) (v) = ~-l(~(g)~(V» on V induced by

~(g) is a diffeomorphism and the Jacobian of f(g) can be calculated

as

* * 1/2 -1 J f (g) (v) = 1 ~ (v) ~ (g) ~ (g) ~ (v) 1 J~ (f (g) (v» •

Furthermore, the geometric measure on ~ is seen to be quasi-invariant

with quasi-mUltiplier

J~ (g) (x)

1 * -1 * -1 11/2 -1 -1 ~ (~ (x»~(g) ~(g)~(~ (x» J~(~ (x» . (6.2)

55

NOw, suppose that ~ has regular orbit type G/K and let (z,u):~ ~

G/K x ~ be an orbital decomposition according to lemma 2.2. Summar­

izing the considerations above we then have that there exists an in­

variant measure u on ~ if and only if. for every u,

J"( (k) (u) 1, k € K. (6.3)

In this case the function

m(x) = J"( (g) (u), g € Z (x) (6.4)

is a modulator with quasi-multiplier (6.2), and hence the measure u is

given by

c4t (x) -1

J"( (g) (u) dA (x) (6.5)

.=i",s,--"i'-'.n....,v,-,a",r=...1",,' a"'n!..!.t""'--"'u""n"'d"'e'-"r'---'t"'h"'e"--a"'-><c->:t.=i->:o"'n'----'''('-----'o'''-f''-------'''G_----''o'"''n-'------''''~ • Note that, by coro 1-

lary 5.1, condition (6.3) is equivalent to

In particular, let ~ = G and consider the left action 0 of G

on itself. This action is transitive and the isotropic group of any

element in G is trivial. The conditions for applying (6.5) are

therefore trivially fulfilled and we find that left invariant measure

a on G exists and is of the form

da (g) -1

J o (g) (e) dA (g).

Similarly we find that right invariant measure ~ is given by

d~ (g) J -1 (e)-ldA (g) e(g )

(6.6)

(6.7)

On comparing (4.3), (6.6) and (6.7) we obtain the following important

formula for calculating the module of G

A (g) -1

J -1 (e)JO ( )(e) • e (g) g

(6.8)

56

Let us introduce an action C = CG - the conjugation diffeomorphism -

of G on itself by the prescription

1

C(g)g

Thus C

A(g)

1 1-1 ggg

E. 00 DOE. and

-1 J C (g) (e) (6.9)

Example 6.1. The

{(Vi)~=1Iv1,···,vn of (IRn)n and

invariant measures on GL(n) and GA(n). Let V

is a basis of IR n }. Then V is an open subset

-/I: V -+ GL(n) n

(vi )i=l -+ {V1 ,···,Vn }

parametrizes GL(n). We have J-/I = 1 so that the geometric measure X

on GL(n) is the restriction of the Lebesgue measure on

open set GL(n). Now let g € GL(n) and let reg) = -/1-1

Then reg) «Vi)~=l) = (gvi)~=l is linear and Jr(g) (v) =

is left invariant. Similarly we have

d~(g) = Igl-n dX(g) ,

showing that GL(n) is unimodular.

(IRn) n to the

o o(g) 0 -/I.

Igl n so that

The general affine group GA(n) = {[g,x] Ig € GL(n),x € IRn} with the

multiplication rule [go,Xo][g,X] = [gog,gox+xo ] is parametrized by

~: V x IR n -+ GA(n)

(v,x) -+ [-/I(v),x]

and the geometric measure is clearly given by dX(g)dx. The mapping -1 n

~ 0 o[g,XO] 0 ~(v,x) = «gvi)i=l,gx+xO) is affine with Jacobian

Igln+l so that

I I-n-l da(g,x) = 9 dX(g)dx

is left invariant.

with the parametrization

~: V x ~n ~ GA(n)

* (v,x) ~ [-/I (v) ,x]

57

--1 -1 it is seen that ~ 0 c([g,Xo] ) 0 ~(v,x)

linear with Jacobian Igl n so that

d~(g,x) = Igl-ndX(g)dx

is right invariant and the module of GA(n) is given by

AGA(n) (g,x) = Igl-1 . D

Example 6.2. The invariant measures on T+lnl. Consider example

4.3 and introduce the following parametrization of T+(n). Let

n

t = k

and define -/I: IT Sk ~ k=1

by T where

{to 0

(T) o. = ~J ~J 0

i ::; j

otherwise.

It is obvious that J == -/I 1 so that the geometric measure on T+ (n) is

given by dX (T) IT dtij · If n E T+(n) and hk = g {gij}i,j=1

i::;j

k then the mapping -/1-1 o 0 (g) -/I(t) n is linear {gij}i,j=1 0 = (hkt k )k=1 n n k

with determinant IT Ihkl IT IT g .. = ~ gn-k+l u k=1 kk

Consequently, the k=1 k=1 i=1

left invariant measure on T+(n) is given by

da (T) n . IT t7-:-n - 1dX(T).

i=1 ~~

In a similar way we obtain the right invariant measure as

djj (T) n . 1I t:-~ dX (T)

i=l l.l.

58

and the module of T+(n) is given by

n 2i-n-1 1I t ..

i=l l.l. o

Example 6.3. GLCn)-invariant measure on PDCn). Consider the action

of GL(n) on PD(n) given by (4.13). It is clear that PD(n) is an

open subset of Sen), the vector space of symmetric n x n matrices,

so that the geometric measure on PD(n) is the restriction d1 of the

Lebesgue measure on S(n). The isotropic group of I € PD(n) is

O(n), and GL(n) = T+(n)O(n) is a left factorization. This means that

if 10 € PD(n),

determinant of the linear isomorphism

f(T O): Sen) .... Sen)

* 1 .... T01To.

we just have to calculate the

The invariant measure is then given by If(T) 1-1d1 where 1 = TT*.

Now, a bit of calculation shows that If(To ) I = IToln+1 = 110 1 (n+1)/2,

so that

is the invariant measure on PD(n).

Alternatively, consider the measure on PD(n) given by

JL (f) * f f(AA )daG(A) GL(n)

where is the (left) Haar measure on GL(n) . To see that this

integral is well-defined suppose that f € ~(PD(n», i.e. f is con­

tinuous with compact support. Then we have to verify that

(A .... f(AA*» € ~(GL(n», but this is obvious since the isotropic

groups are compact. Furthermore, it is easy to see that JL is invari­

ant i.e. JL = JL. Now T+(n) x O(n) acts transitively and freely on

GL(n) by the prescription

* ~(T,U): A ~ TAU.

59

By Example 6.1, u G is both left and right Haar measure, implying that

is also invariant under ~ and the same holds for the measure

given by

~(f) = * J J f(TU )dUduT (T) T+(n) O(n) +

where dU is the (left) Haar measure on O(n) and is the left

This J * Haar measure on T+(n) . shows that Jl (f) f(AA ) dUG (A) = GL(n)

J * * * J f(TU UT )dUduT (T) J f(TT )duT (T) if dU is norma-T+ (n) O(n) + T+ (n) +

lized to be a probability measure on O(n).

Consequently, we obtain that

* Jl (f) = J f(TT )duT (T) (6.11) T+ (n) +

is another way of representing the invariant measure on PD(n). D

Example 6.4. The Blaschke-Petkantschin formula. In continuation of

example 2.8 and example 4.6 let dX denote the restriction to

gl(p,n)t of Lebesgue measure on gl(p,n) and consider the action of t G = GL(p) x O(n) on gl(p,n) given by

gl(p,n)t G x 91(p,n)t ~ «A, U) ,X) * ~ AXU

Then dX is relatively invariant with multiplier K(A,U) = Idet(A) In,

as follows from observing that ~(A,U) = 8(A) 0 c(U) and that 8(A),

c(U) are linear. Secondly, we have in analogy with the considerations

in example 6.1 that

det(~(A,U)) = det(8(A))det(c(U)) = det(A)n det(U)P = det(A)n

On the other hand, let IAI-PdA denote the invariant measure on

GL(p) and let dL denote the invariant measure on G(p,n) =

t GL(p)\gl(p,n) .

60

Now, define a measure ~ t on gl(p,n)

~(f) J J f(AX) lAin IAI-P dA dL . G(p,n) GL(p)

by

It is easily verified that ~ is also relatively invariant with multi­

plier x. with a suitable normalization it follows that we have

J f(X) dX J J f(AX) IAl n - p dAdL

gl(p,n) t G(p,n) GL(p)

This is known as the Blaschke-Petkantschin formula (Blaschke 1935a,

1935b and Petkantschin 1936), and is a useful tool in stochastic geo­

metry, e.g. in proving unbiasedness of stereological estimators (see,

for instance, Miles, 1979, Jensen and Gundersen 1988, and Jensen, Kieu

and Gundersen, 1988). o

Example 6.5. Invariance subgroups of GL(n). Let ~ E GL(n) and

let H be a closed subgroup of G = (g E GL(n) Ig~g*=~}. Suppose that

~:V

2 .... IR n

chart on

E GL(n).

-1 is a local parametrization such that ~ = ~ provides a

H, and let Lg (Rg) denote left (right) translation by g

2 Then Lg and Rg are linear isomorphisms of IRn. In order

to determine a left invariant measure on H, we must calculate the

Jacobian at the identity of the diffeomorphism

-1 -1 f(Lg) = ~ 0 Lg 0 ~: V .... V, ~ (g) E V.

Now, and with -1

~ (e) , we obtain the follow-

ing equation for the differentials

This implies that

61

* ~ ~ ~ (e) 0 L -1 0 R~ 0 ~(e),

~

where the last equality follows from the fact that L * 0 L -1 0 Lg g ~

L -1. ~

Hence, taking determinants it follows that

where the constant c is given by

Consequently, a left invariant measure on H is given by

similarly, replacing Lg by Rg and observing that R * 0 R~ 0 Rg g

R~, it follows that a is also right invariant, showing that H is

unimodular.

In particular, this example applies in case H is a closed subgroup

of O(n) or O(1,q) or O(p,q). 0

Example 6.6. Orbits of invariance subgroups. Let H be a group of

the type considered in the previous example and let Xo € mn. Then we

want to determine an invariant measure on ~ = {gxolg € H}, i.e. on

the orbit of

Suppose that is a parametrization of ~. Then

arguments similar to those given in example 6.5 show that

(6.13)

is an invariant measure on ~. We consider three special cases.

(i) Next, consider example 3.5 and the parametrization of

62

given by the polar coordinates

"'1 (9) R(9)XO' 9 € 9 , (6.14)

where R(9) is a sequence of rotations. Some tedious calculations

yield

J(9)

so that the invariant measure on Sp-l, which is also the geometric

measure, is given by

dM 1 (9) = J(9)d9. (6.15)

(ii) Reconsider example 3.6 and the parametrization of Hq given by

U € IR, (6.16)

where 'l'(u,t) = chu + 1/2 e U lltll2 and ~(u,t) = shu + 1/2 e Ulltll2.

Performing the requisite calculations, we obtain

{-I-lItIl 2

D>/I;(U,t)II D>/I2(U,t) = ,q -t

* } -t

-I q-l

which has determinant (-I)q so that the invariant measure on Hq is

given by

dudt1 ... dtq _1 (6.17)

(iii) Consider example 3.7 and the parametrization of HP,q (except

for a null set) given by

63

[

'T (u,tHl (8)] ~(8,u,t) = ~(:,t) (6.18)

Calculations yield

* ~ (8,u,t)I ~(8,u,t) p,q

which has determinant (_1)qJ2 'T 2 (p-l), so that the invariant measure

on HP,q can be represented as

dj.!(8,u,t) = (chu + 1/2 e U lltll2) (p-l)J(8)d8dudt. (6.19)

An alternative parametrization of HP,q is given by

;j;(8,s) (6.20)

where /3(s) (1 + IIsIl2)1/2, and we obtain

-2 * J /3 ss-I

with determinant (_1)qJ2/32(p-2), so that another representation of

the invariant measure is given by

dj.!(8,s) = (1 + IIsIl 2 )1/2(p-2)J(8)d8ds. (6.21)

o

Example 6.7. sol (l,gj-invariant measure on cone surface. Consider 1 the action of so (l,q), q ~ 2 on the orbit

!'r o o}

64

In this case the methodology of the previous example breaks down, be­

cause the determinant in (6.13) is identically zero. Instead we use

(6.5) to construct an invariant measure. This requires an orbital de­

composition, which may be constructed via Lie algebras along the lines

in section 3, leading to the conclusion that

i.e. ~o is isomorphic to a generalized hyperboloid.

However, the decomposition is almost trivial to establish in the

following way.

We consider the parametrization

"'(x) = ["~"] ,

and define the decomposition by

"'(x) = g(x)uo

where

e~ (1,0, •• ,0) € mq ,

g(x) o } {Z(X) U(x) 0

{~(IIXII+IIXIl-1) ~(IIXII-IIXIl-1)} z(x)

~(IIXII-IIXIl-1) ~(IIXII+IIXIl-l)

and U(x) € O(q) is a sequence of rotations determined by

since

x II xII

65

it is easy to see that

Applying (6.5) it follows that

dJ,t(x) = IIxll- l dx

is the invariant measure on ~O'

For later purposes we note that ~o is also invariant under the

action of ~: x sol (l,q) given by

(a ,A) : [ II XxII]

and it is obvious that ~ is relatively invariant with multiplier -_ a q - l . x(a,A) 0

We close this section by an example illustrating theorem 5.5, which

characterizes the quotient measure by invariance.

*. . . Example 6.S. SO(p,g) x ~+-~nvar~nt measure on a cone. Cons~der

example 5.2 and the action of * SO(p,q) x ~+

and xl > 0 if p = I}, given by

The mapping

(s, r) :

transforms the invariant measure

respectively p, is invariant on

r(x)-(p+q) dx into

HP,q, respectively

where a,

66

The invariant measure on is given by -1 r dr

f f(x)r(x)-(P+q)dx !I

CX>

f f f(r- 1s)r-1drda(s) HP,q 0

or equivalently, applying (6.21),

f f(x)dx !I

so that

In case q = 0 this is the well-known polar decomposition of Lebesgue

measure on ffiP. 0

Bibliographical notes

This section consists primarily of examples involving the notion of

differentiable manifolds and Lie groups. A comprehensive, but rather

succinct, exposition of these concepts are given by Helgason (1978),

chapters I and II, while Cohn (1957) provides a very nice introduction

to Lie groups.

7. Exterior calculus

The exterior calculus of differential geometry provides procedures

for factorization of measures and for the construction of invariant

measures, which in many cases constitute a shortcut to the result. We

wish here to indicate the technique so as to enable the reader to apply

it without having to study exterior calculus as such. Accordingly, the

discussion will in the present section be somewhat informal in compari­

son with the previous sections. For a comprehensive and rigorous expo­

sition of exterior calculus see, for instance, Edelen (1985).

Exterior calculus can be said to be the calculus of differentials.

We shall start by illustrating, through an example, how manipulations

with differentials can sometimes, in a simple and elegant way, lead to

a desired factorization of a measure. Actually, in the example, we use

- except for a reference to the result (5.11) - little more than stand­

ard reasoning of ordinary calculus.

67

Example 7.1. A factorization of Lebesgue measure on PD(n). Let

(7.1)

be Lebesgue measure on the set

matrices viewed as a subset of

PD(n) of positive definite n x n

Rn (n+1)/2. We seek a factorization of

~ corresponding to the spectral decomposition

* UAU (7.2)

of an arbitrary matrix ~ € PD(n), where A is the diagonal matrix of

eigenvalues of ~ and U is an element of O(n). Disregarding the

nullset N of PD(n) corresponding to multiple characteristic roots

we may assume that the diagonal elements of A satisfy 0 < All < A22

< ••• < A nn The set of all such A will be denoted by ~, and we

shall write Ai for Aii . There are (n-1)n/2 functionally indepen-

dent elements of U and we choose to work with

n as such a set of elements.

The group G = O(n) acts on PD(n) by the law

O(n) x PD(n) ~ PD(n)

* ~ U~U

for

Under this action the set ~ = PD(n),N is invariant and of constant

orbit type, the matrix A is a maximal invariant and the isotropy

group at A consists of the set of diagonal matrices whose diagonal

elements are +1 or -1. Furthermore, the action is proper and the

measure (7.1) is invariant under the action. Hence, by theorem 5.2, we

have that ~ factorizes in the sense that for any integrable f

* f f f(UAU )d~(U)dK(A) ~ G

where ~ is right invariant measure on O(n).

The mapping

G x ~

(U,A) * ~ UAU

(7.3)

68

is locally a diffeomorphism and consequently K

with respect to Lebesgue measure rrd~i on ~.

has a density h(A)

To find h(A) we first

note that (7.2) entails

* * * dUAU + UdAU + UAdU (7.4)

where cu: , dU and dA

differentials of ~, U

are the

and A,

n x n matrices consisting of the

* respectively. since UU = I we have

* * dUU + UdU o and hence

* * * UdAU + dUU ~ - ~dUU (7.5)

from which we may determine an expression for rri~j da ij and conse-

quently the form of h(A). In view of the factorization (7.3) it suf­

fices to consider the case U = I, when

d~ dA + dUA - AdU

or, expressed in terms of the entries of the matrices,

where

h(A)

is the Kronecker delta. Consequently,

rr (~. -A . ) • i<j J 1

Let M be an m-dimensional differentiable manifold and 1 m be local coordinates M. with each point w , ••• , (IJ on p

associate an m-dimensional vector space * TMp (the dual of

space TMp of M at p) and a particular basis of TM * p'

D

let

of M we

the tangent

the m

vectors of the basis being denoted by dw 1 , ... ,dwm. The details need

not concern us here, what is important in the present context is how

the basis changes under a change of coordinates. Suppose ~l, ... ,~m is a set of alternative coordinates, and let us denote generic

coordinates of w = (w 1 , ... ,wm) and ~ = (~l, ... ,~m) by wr , wS , ••.

and ~a, ~b, ... , respectively. Furthermore, if f is a dif­

ferentiable function on M we write fir for 8f(w)/8w r , and in

particular we then have ~;r 8~a/8wr. The transformation rule for

69

the basis vectors may now be stated as

d ."a __ .I,a d r ." ." /r w • (7.6)

Here, and in the sequel, we employ the Einstein summation convention.

Note that this transformation rule fits in with the standard (mostly

informal) way of expressing the differential df of a function f as

df(w) r fir dw (7.7)

for if on the right hand side of the expression df(",) = f d",a /a we

insert (7.7) there results Le. df(w).

It is possible to introduce a product of vectors in termed

the exterior product or the wedge product. In particular, the exterior r l rm

product of m basis vectors dw , ... ,dw in that order, is denoted

by

m r, A dw 1

i=l

r l r dw A ... Adw m

Again, most of the details need not concern us here, but it

tial to know that the wedge product has the properties that

more of the indices rl, ... ,rm coincide then (7.8) equals

and if o(l), ... ,o(m) is a permutation of l, ... ,m then

(7.8)

is essen-

if two or

0 Le.

(7.9)

(7.10)

Using these rules and the distributive property of the wedge product we

obtain from (7.6) that

70

where J~(w) is the Jacobian determinant of the transformation from w

to ~.

As discussed in section 6, the determination of an invariant measure

can usually be reduced to the calculation of a Jacobian. The point we

want to make here is that the above-mentioned properties of exterior

multiplication are convenient for such calculations. Example 7.1 can,

in fact, be seen as illustrating this. As another instance we consider

Example 7.2. Jacobian of the Cholesky decomposition. Consider the

one-to-one mapping of T+(n) onto PD(n) given by T ~ I where

* T T

is the (unique) Cholesky decomposition of

Denoting the entries of T and ~ by

ly, we have

and hence

i I (tk,dtk,+tk,dtk ,).

k=1 ] ~ ~ ]

and respective-

(7.12)

To obtain the Jacobian JI(T) of the transformation from T to I we

calculate the exterior product A da l,],. In view of (7.9), when dt~], i~j ~

has occurred in one of the factors we may disregard it in any

subsequent factor. Bearing this in mind we write out (7.12) as

da1n

dann

from which it follows that

A da 1· J' i~j

so that

J~(T) n II

i=l t~:-i+1.

11

71

Let 1 ~ r ~ m. A linear combination of the type

o

(7.13)

where the functions f. . are smooth and we are using the Einstein 1 1 ... 1 r

summation convention, is called a (differential) r-form on M. Without

loss of generality we assume that is skew-symmetric, i.e.

f. . 1a(1)·· .1a (r)

sign(a) f. . 1 1 ... 1 r

for any permutation a(l), ... ,a(r) of 1, ... ,r. Note that it is im­

plicit in (7.13) that for any given coordinate system on M we have a

collection of smooth functions and that, in an obvious notation, these

are related by

(7.14)

(In the language of differential geometry this means that

a covariant tensor of rank r.)

72

A differential r-form (7.13) is said to be invariant under the ac­

tion of a Lie group G on the manifold M if for every fixed g E G

and with the coordinate system ~ defined by ~ = gw, i.e. ~(p)

W (g-lp) f . t E M h or any pOln p , we ave

f. . (gw) 1 1 ···1r

(7.15 )

where, in this context, 6 is the Kronecker delta.

We conclude this section by exemplifying how the invariant measure

on an m-dimensional differentiable manifold M under a transitive

action of a Lie group G may be determined as the exterior product of

m invariant I-forms.

In general, suppose that M is a submanifold of ffin and that G

is a closed subgroup of GL(n). Let K be the isotropic group of Xo

E M, and suppose that 4 K(k) 4G(k), k E K, i.e. there exists an

invariant measure on M (cf. corollary 4.2). Furthermore, we will

assume that G = HK is a left factorization of G with respect to K,

so that M = HXO. Then for x = hxo' h E H, we have that h-1 dhxo * (w 1 (x), ... ,wn (x» is a vector of invariant I-forms on M. If we

choose m linearly independent I-forms ~1' ... '~m among w1 , ••• ,wn

then the exterior product

d~ (x)

of determines an invariant measure on M.

Example 7.3. The invariant measure on HP,q. According to example

2.6 that HP,q is parametrized by

Q(9,y,J,L,t)uO

where x = ~1(9,y) E Sp-l and (~) = ~2(J,L,t) E Hq . For short, let Q

Q(9,y,J,L,t). since Q = R(9,y)P(J,L,t) = RP it follows that

NOw, let

73

:} . Then i,j

* (w 1 ,···,w 1 ) P p- ,p

1, ... ,p, are invariant 1-forms on sp-1 and w*p =

is a vector of linearly independent 1-forms deter-

mining the invariant measure on Sp-1 given by

dw(x) = w 1 (X)A ... Aw 1 (x) P p- ,p

Similarly, if

{:

then determines an invariant measure on Hq •

combining the above we obtain

-1 Q dQuo [aw*p 1 2 a w + PO~

awp:~ + P*O

where

[::::v + ..J is a set of linearly independent, invariant 1-forms on HP,q.

w A ... Aw Aw = 0 we have that the invariant measure on 1p p-1,p pp

the point Quo = (~x) is given by

p-1 a dw(x)Adp(a,v)

Since HP,q at

74

where dw is the invariant measure on Sp-1 and dp is invariant on

Hq • This is in agreement with the result obtained by combining (6.15),

( 6 . 17 ) and ( 6 . 19) • 0

Bibliographical notes

For more details and further examples concerning the topics dis-I

cussed in this section, see Santalo (1979) part II, Muirhead (1982)

chapter 2, and Farrell (1985).

8. statistical transformation models

Many of the most important models in statistics are transformation

models, or partly transformational. In this chapter we present the key

aspects of the powerful theory of transformation models. This theory

draws heavily on the theory of decomposition and invariance of measures

considered in sections 2-7.

To make the account accessible without a detailed knowledge of the

material in the previous sections, we have chosen to formulate the

definition and the main theorems of transformation models as non-tech­

nical as possible and to discuss the more technical aspects, including

the appropriate regularity conditions, in the form of comments. We do

assume, however, that the reader is familiar with the basic notions

relating to group actions and to invariant measures, as given in the

beginning of sections 2 and 4, respectively. Furthermore, a good know­

ledge of parametric statistics is essential for a full appreciation of

the import of the material in this section.

We say that a class of probability measures ~ defined on a sample

space ~ constitutes a transformation model with acting group G if

G acts on ~ and if there exists an element P of ~ such that ~

{gP: g € G}. Here gP denotes the measure P lifted by the transfor­

mation corresponding to g € G, i.e.

(gP) (A) (8.1)

for every A € ~, the class of measurable sets. Note that, more gener­

ally, an action of G on ~ induces, via formula (8.1), an action of

G on the set of all probability measures on ~. The equality ~

{gP: g € G} says that ~ is invariant under this induced action and

that ~ consists of a single orbit. If we require only that ~ be

75

invariant relative to G we have the concept of a composite transform­

ation model with acting group G.

In the present section we consider primarily models ~ which are

pure transformation models, but some discussion of composite transform­

ation models is provided at the end of the section. After some first

examples of transformation models we formulate the concept of a stan­

dard transformation model. Virtually all transformation models met in

statistics are of this type, and for such models we then state and

prove a theorem which is the main tool for statistical inference under

transformation models. The proof of this result (Theorem 8.1) relies

strongly on the theory developed in sections 2-7, but in line with our

introductory remark we have endeavoured to formulate the theorem so as

to make it comprehensible and applicable without a primary close atten­

tion to the requisite technicalities and regularity conditions. A con­

siderable diversity of examples is subsequently discussed, to illus­

trate the versatility and power of this theorem. The concludig discus­

sion of composite transformation models contains as a main result a

formula for the marginal likelihood based on the maximal invariant

statistic.

As an incidental comment it may be noted that most of the models

considered in the examples of the present section are exponential

transformation models, i.e. they are both transformation models and of

the exponential family type. Such models thus possess two quite differ­

ent sets of general, theoretically and practically significant, proper­

ties. They are therefore of particular interest and they have various

special structures of import. These structures are indicated in exer­

cise 25 (p. 128). A brief exposition of the structures may also be

found in section 2.5 of Barndorff-Nielsen (1988), where references to

the literature are given.

Example 8.1. Location-scale model. Let G be the representation

of the location-scale group considered in example 4.9, i.e. G is the

group of 2 x 2 matrices

g [~ ~] a > 0, f € IR •

The location-scale group acts on IR n by the law

G x IRn -+ IR n

(g,(x1 ,···,xn » -+ (f+axl,···,f+axn) (8.2)

76

Let f: m ~ m+ denote an arbitrary probability density function

(with respect to Lebesgue measure on rn) and let P be the probabili­

ty measure on mn corresponding to the assumption that the variables

x 1 , ... ,xn are stochastically independent and identically distributed

each with probability density function f. Then

~ cD. (X1 ,···,X) n n

n a-n IT

i=1

x·-f f(-~-)

a

where An denotes Lebesgue measure on rnn.

(8.3)

The model ~ = {gP: g € G} is a transformation model, called the

location-scale model corresponding to f. o

Example 8.2. Additive effects model. (Multivariate two-way analysis

of variance). Let x .. i=1, ... ,k, j=l, ... ,l, v=l, ... ,m, be indepen­~]v

dent, p-dimensional random vectors with x .. ~]v

following a mUltivariate

normal distribution. We shall discuss the model for (fixed) additive

effects, that is x .. ~]v

is assumed to have mean vector of the form

+ ~j + ~ and covariance matrix ~, where the p-dimensional vectors

a i (i=1, ... ,k), ~j (j=1, ... ,I) and ~ and the symmetric positive

definite matrix ~ are all considered as unknown parameters. We let n

k-I-m and q = k-I-m-p.

Define G to be that subgroup of GA+ (q) whose typical element g

[B,f] has f of the form f ijv = a.+~.+~ :1 ]

and B of the form B

A~In with In the n x n identity matrix and A an arbitrary ele-

ment of GL+ (p) . In order to identify the effects we assume ~a. = 0 ~

The additive effects model is a transformation model under the

action of G on inherited from the action of as defined

in example 2.1. If we let P denote the probability measure on mq

corresponding to the standard normal distribution Nq(O,I) then gP

with ~ = AA *. o

Example 8.3. von Mises-Fisher model. The von Mises-Fisher distri­

bution on Sp-1 = {x € rn P : IIxll=1} (or the circular normal distribution

in mP ) with mean direction f € Sp-1 and concentration parameter K

~ 0, which we denote by Cp(f,K), has probability density function

77

with respect to the surface measure ~ on of the form

X € sp-1 . (8.4)

Here indicates the usual inner product in mP and ap(K) is a

norming constant given by

p/2 1-p/2 (2v) K I p/ 2_1 (K)

where denotes the modified Bessel function of the first kind

and of order p/2-1.

As mentioned in example 3.5, the special orthogonal group SO(p)

{U € GL(p) Idet(U) = 1, uu* = Ip} acts on Sp-1 by the law

SO(p) x Sp-1 ~ Sp-1

(U,x) ~ Ux .

Since the measure ~ is SO(p)-invariant one has that

(8.5)

(where X denotes a random variate and is read "distributed as").

The sample space ~ corresponding to a sample of size n from the

von Mises-Fisher distribution is

and SO(p) acts on ~ by the law

SO(p) x ~ -+ ~

(U, (x 1 ' ... , Xn » ~ (Ux 1 ' ... , UXn ) . (8.6)

The measure on ~ is invariant under this action. If

denotes the probability measure on ~ corresponding to a sample

x 1 , •.• ,xn from the Cp(f,K)-distribution it follows from (8.4) that

dP (f ,K) ~ (X1 ,···,x)

dj.L n n (8.7)

78

where x+ x 1+ ... +xn . Furthermore, (8.5) implies that

for U € SO(p) (8.8)

The action of SO(p) on sp-1 is transitive, implying that for every

fixed K > 0 the class of probability measures

constitutes a pure transformation model, whereas the class

_ I p-1 f - {P(f,K) f€S , K~O}

constitutes a composite transformation model. o

Example 8.4. Elliptical models. Consider the action of GA(p) on

mP given by

([A,a],x) ~ Ax + a .

It is well-known that the class of (regular) mUltivariate normal

distributions is invariant under this action and constitutes an example

of a transformation model, i.e. the action is transitive. If we con­

sider the parameterization Hp = {Np (f,2) If € mP ,2 € PD(p)}, where f

is the mean vector and 2 is the covariance matrix, then the induced

action of GA(p) on mP x PD(p) is given by

([A,a], (f,I» * ~ (Af+a, AlA )

The isotropic group of (0,1) equals O(p), i.e.

Hp - GA(p)/O(p) .

NOW, suppose that P is a probability measure on mP having densi­

ty with respect to some quasi-invariant measure. It is then a natural

question to ask when the model ~ = (gPlg € GA(p)} can be parametrized

by GA(p)/O(p). This problem has been solved by Jespersen (1989) who

shows that ~ has to be elliptical, i.e. P has density f with

79

respect to Lebesgue measure on ffiP of the form

x € ffiP ,

where

h: ffi+ ~ ffi+ is continuous

and

sm h(s) sp/2-1 ds o f(P/2)'IT-P/ 2 .

It follows that

f(f,};) (x)

where }; * AA

d([A,f]P) (x) dx

In terms of densities we have

If X ~ f (ft};) and S~ h (s) sp/2 ds < 00, then the mean value

is f whereas the covariance matrix of X is proportional to

class of mUltivariate normal distributions is determined by

h(s) = (2'IT)-p/2 exp(-s/2) ,

while

h(s) = (n'IT)-P/2f «n+p)/2)/f(n/2) (1+s/n)-(p+n)/2

of X

L The

defines the class of mUltivariate student distributions with n de­

grees of freedom. When n = 1 these are called the multivariate Cauchy

distributions. 0

Before we give the definition of a standard transformation model we

recapitulate the necessary concepts concerning the action of a group G

on a space ~. An orbit representative u is a maximal invariant

function defined on ~ and with values in ~, i.e. u(gx) = u(x) E ~

80

for x € ~ and 9 € G and u(x) = u(x') if.and only if there exists

a 9 € G such that x' gx.

relative to the action of G

u and a regular subgroup K

u(x)} = K for every x € ~.

The space ~ is of regular orbit type

if there exists an orbit representative

of G such that GU(X) = (g€Glgu(x)

In that case we refer to u and K as

the orbit regresentative and the isotrogy groug associated to G. A

mapping t defined on ~ and with range space ~ is

t(x) = t(x') implies. t(gx) = t(gx' ) for every 9 €

acts on the range

scription

Gx~-+~

(g,t) -+ gt

where

gt t(gx)

space

if t

We are now set for

~ of an equivariant mapping

t(x) •

egyivariant if

G. The group

t by the pre-

~,

G

Definition 8.1. A transformation model ~ with sample space

acting group G and probability measure family ~ = {gP: g€G} is said

to be a standard transformation model if the following conditions are

satisfied:

(i) There exists an invariant measure ~ on ~ such that

P «~ (i.e. P

~). We let p

Le. p = dP/~.

is absolutely continuous with respect to

denote the density of P with respect to

(ii) ~ is of regular orbit type relative to the action of G.

~,

The orbit representative and the isotropy group associated to

G will be denoted by u and K, respectively.

(iii) K is a subgroup of K, where K Gp = {g€GlgP=p}.

(iv) There exist equivariant statistics rand s and a statistic

v with range spaces ~, ~ and 1, respectively, such that:

(a)

81

G acts transitively on ~ and on !I,

the range space of u, one has Gr(u)

K, so that the actions of G on ~

identified with the natural actions of

G/K and G/K, respectively.

and for

= K and

and !I may

G on the

(b) ~ is in one-to-one correspondence with K/K.

u € 11,

Gs(u)

be

spaces

(c) There exist measures p, a, u and ~ on ~, !I, 11

and ~, respectively, such that under the bijections

!: .... ~xll !: .... !I x 11 x l'

x .... (r,u) x .... (s,u,v)

the invariant measure ~ is lifted as

~ .... p at u

where p and a are invariant measures under the action

of G on ~ and !I, respectively. []

comments

(A) As stated in theorem 8.1 below, condition (i) of definition 8.1

implies that the transformed probability measure gP has a density,

which will be denoted by gp, with respect to the invariant measure.

This density is of the form

~ -1 gp(x) = ~ (x) = p(g x) , X € ~ •

Consequently, by condition (iii), an equivalent characterization of the

subgroup K is given by

K (g € Glp(g-l o) = p(o) <~>} (8.9)

(Here <~> indicates that the equality holds except possibly for a set

of ~ measure 0.)

(B) On account of (8.9), there exists a one-to-one correspondence

between the set , of probability measures and the set of left cosets

G/K = {gKlg € G}, i.e. the transformation model , may be para­

metrized by G/K. However, such a parametrization is often considered

82

as artificial and we have here chosen to use the space ~ as a para­

meter set. According to (a) in condition (iv) the set ~ may be

thought of as a representation of the space of cosets G/K and in that

case the statistic s is typically the maximum likelihood estimator.

In applications one often has a left factorization G = HK of G with

respect to K, as discussed in sections 2 and 3, and in that case ~

is simply a representation of the subset H of G.

(C) For a transformation model satisfying that the maximum likeli-

hood estimate sex) exists uniquely for every x € 3: and for which

the orbit representative u in (ii) is chosen such that G = s(u) K for

every u € 'tI, it is easily seen that K C K. This observation may be

viewed as the justification for condition (iii) in definition 8.1.

Models with K = K are special instances of the socalled balanced

transformation models, introduced and studied in Barndorff-Nielsen

(1983, 1988). Note that if K = K the statistic v in condition (iv)

is trivial. The location-scale model with sample size n ~ 2 is ba­

lanced, but for n = 1 we have K J K (cf. the continuation of ex­

ample 8.1 below).

(D) From unbalanced transformation models, in particular unbalanced

exponential transformation models, it is sometimes possible by margina­

lization to a minimal sufficient statistic or by conditioning on an

ancillary statistic to obtain balanced transformation models (cf. the

continuations of example 8.3 below). According to the statistical prin­

ciples of sUfficiency and ancillarity a model derived in one of these

ways from an original transformation model is an appropriate basis for

inference on the model parameters.

(E) In example 8.4 and example 8.6 below the action of G on the

sample space is free, i.e. K = {e}, but the transformation model is

unbalanced. However, in both examples there exists a subgroup Go S G

generating the same transformation model as G, but such that the

model corresponding to Go is balanced.

(F) Finally, note that the assumption Gr(U) = K in (a) of condi­

tion (iv) implies that the mapping x ~ (r,u) is, in fact, a bijec­

tion. Furthermore, according to (a) and (b) we have the identifications

~ - G/K - G/K x K/K - ~ x 1

and hence the mapping x ~ (u,s,v) is also a bijection, and it corre-

83

sponds to the identification

II == II/G x G/K x K/i{ D

until now no regularity conditions have been mentioned in relation

to a standard transformation model. These conditions, which are all of

topological nature, are as follows:

(i) (G,II) is a standard transformation group (cf. p. 12).

(ii) The spaces ~, ~ and 1 are LCD-spaces (cf. p. 12) and

(G,~) and (G,~) are standard transformation groups.

(iii) The mappings x ~ (r,u) and x ~ (s,u,v) are proper mappings

(cf. p. 13).

Remarks. Note that endowing

(i) implies that G acts on ~

G x ~ ~ ~

(g,P) ~ gP ,

where gP is given by (8.1).

~ with the weak topology, condition

according to the law (cf. p. 3)

In all examples of non-trivial standard transformation models we

know of, the isotropy group K is compact, which by (iii) in defini­

tion 8.1 implies that K is compact. We have not been able to prove

that in general K is compact: however, this is certainly the case if

the action of G on ~ is proper, cf. (iv) on p. 13.

Example 8.1. Location-scale model (continued). According to example

4.9, if n = 1 there exists no invariant measure on m under the

action (8.2) so in that case the location-scale model is not standard.

From the same example it follows that if we let K = G(O,l)*' the

isotropy group corresponding to x = 0, one has that

K {[~ ~]: a > O} .

For n ~ 2 let

84

- 1 n

where x }; xi· Restricting the sample space to ~ = { x€lRn : n i=l

s(x»O}, it follows from example 4.1 that the measure on ~ given by

dj.L(x) -n s (x) dAn (x) (8.10)

is invariant under the action of the location-scale group as defined by

(8.2). From (8.3) and (8.10) one obtains that

p(x) dP(X) dj.L

n s (x) n rr f (xi) .

i=l (8.11)

As shown in example 2.3 the action is free, i.e. K consists of the

2 x 2 unity matrix I2 alone, and the 'configuration statistic', i.e.

- -u (x)

x -x x -x 1 n

(--s-' ... '--s-) (8.12)

is an orbit representative and is an ancillary statistic. Furthermore,

the mapping

r: ~ -+ IR+ x IR

x -+ (s(x),x)

defines an equivariant statistic such that r(u(x» = (1,0) for all x

€ ~ and it follows from example 2.3 that the mapping x -+ (r,u) is

one-to-one and onto.

since f is assumed to be the probability density function of a

non-degenerate distribution one has that G acts freely on

gP = g'P if and only if g = g'. Consequently K = {I2 }.

g = [~ i) € G

'!I, Le.

Identifying

with (a,f) € IR+ x IR it follows that ~

action of G on ~ is

IR+ x IR and that the

G x (IR+ x IR) -+ IR+ x IR

(g, (s,t» -+ (as,at+f) (8.13 )

85

or, equivalently, the left action of G on

obtain that the invariant measure p on ~

-2 dp(s,t) = s dsdt.

G. From example 4.9 we

is given by

(8.14)

According to example 5.3 the measure u turns out to be the uniform n

measure on the set {x € IRnl}; xi = 0, IIxli = I} which may be identi­i=l

fied with n-2 S •

The above considerations show that the location-scale model is a

standard transformation model if n ~ 2. o

Example 8.2. Additive effects model (continued). An invariant

measure on IRq under the action of the group G, previously defined in

this example, exists if and only if f > p, where f = n-(k+l-l) is

the 'number of degrees of freedom'. Such a measure is constructed

below. The condition f > p, which we henceforth assume to be satis­

fied, is also necessary and sufficient for the existence ~n~ ~n~queness

with probability 1 of the maximum likelihood estimator (a,~,7,};) of

(a,~,7,};), where we have set a = (a1,···,ak ) and ~ = (~l""'~l)'

Let

- 1 }; xi++ 1m x .. j,v ~Jv

- 1 }; x+ j + km x ..

i,v ~Jv

x+++ 1 n

}; x .. i,j,v ~Jv

and

Defining 3: by

3: = {X€lRq: z positive definite} ,

we have that 3: is an open and invariant subset of IRq and that the

86

probability mass of ~ is 1. Moreover, on ~,

- -a. 1 xi++-x+++, {jj 'Y =

and

1 - z . n

Lebesgue measure X on ffiq is relatively invariant under G with

multiplier ~(g) = lAin which, since ~ = AA*, may be reexpressed as

~(g) = 1~ln/2. Further, one sees immediately that lil n/ 2 is a modu­

lator with multiplier ~(g). Consequently, by (4.2), the measure

c4L (x) = I i (x) l-n/ 2 dX (x) (8.15 )

is an invariant measure on ~, and the corresponding model function of

the additive model is

p(x;a,{j,'Y,~)

(8.16)

where all multiplicative constants have been absorbed in ~.

The group G acts on the range space ~ of the maximum likelihood

estimator

s (a,{j,'Y,~) (8.17)

by

gs (8.18 )

Under this action, Lebesgue measure X~ on ~ is relatively invariant

87

with multiplier IAlk+l+p. In fact, it follows from (8.18) that the multiplier is IAI (k-1)+(1-1)+1+(p+1) IAl k+l +p . Consequently invari-

ant measure on the range space ~ of s is given by

da(s) = lil-(k+l+p)/2 dA~(s) . (8.19)

In view of these properties one sees that the additive effects model

is a standard transformation model with K = {e}, K = SO(p) and with

s, ~ and a given by (8.17), (8.15) and (8.19). o

Example 8.3. von Mises-Fisher model (continued). For simplicity we

restrict ourselves to the case p = 3, i.e. to the Fisher distribution

on the sphere and furthermore we assume that K is fixed. In order to

establish that this model is a standard transformation model, note

first that (8.7) implies that (i) in definition 8.1 is fulfilled. Let-

* ting P be the Fisher distribution with mean direction (0,0,1) it

follows from example 3.5 and formula (8.8) that

K = {{~ ~}: U € SO(2)} ,

i.e. K is the set of rotations in the 1-2 plane.

Since the action of SO(3) on s2 is transitive we may if the

sample size n equals 1 consider

tive and so, in this case, one has

Now suppose that n ~ 2 and let

* (0,0,1)

K = K.

as an orbit representa-

where x+ = x 1+ ... +xn . Clearly ~ is invariant under the action (8.6)

and ~~n is an invariant measure on ~. In order to find an orbit

representative we now, inspired by the considerations in example 3.5,

construct a mapping z: ~ ~ G such that

-1 z(Ux) = z(x)U for U € SO(3) ,

by the following considerations. Let x € ~ and let x+

If we let

(8.20)

otherwise

we define R23 (X) as the uniquely determined rotation in the 2-3

88

plane, such that

similarly we determine R13 (x) as the unique rotation in the 1-3 plane

such that the corresponding angle lies in the interval [-v/2,v/2] and

such that

(8.21)

Finally, we let i = 1,2, ... ,n and j = min{i:

(Xi1,Xi2)~(O,O)} and determine R12 (X) as the unique rotation in the

1-2 plan such that

The function z: ~ ~ G defined by

z (x) (8.22)

satisfies (8.20) and it follows that u: ~ ~ ~ given by

u(x) = z(x)x , X E ~ ,

is an orbit representative and that r: ~ ~ G given by

-1 rex) = z (x) , x E ~ , (8.23)

is equivariant. Consequently, one has that x ~ (r(x),u(x» is an

orbital decomposition, i.e.

x r(x)u(x) . (8.24)

The action of G on ~ is free. In order to prove this it suffices

* For u E ~ one has that u+ = (O,O,lIu+lI)

and that there exists a j such that (Uj1 'Uj2 ) ~ (0,0). The condi­

tion

89

Uu u

implies in particular that

(8.25)

and

(8.26)

From (8.25) one obtains that U must be a rotation in the 1-2 plane

and (8.26) then implies that U = 1 3 . since K = {1 3 } we have that K

~ K and G/K = SO(3). Thus setting ~ = SO(3) and letting r be the

mapping defined by (8.22) and (8.23), the part of (iv) (a) in definition

8.1 pertaining to the statistic r is fulfilled. Furthermore, letting

~ = s2 we define the equivariant mapping s in definition 8.1 by

Note that (8.21) implies that

sex)

Finally, since K/K

v: ~ -+ K -1 x -+ R12 (X)

K we set .,

(8.27)

K and define v by

Noticing that the SO(3)-invariant measures p and a on ~ and ~

are, respectively, the invariant measure on SO(3) and the surface

measure on S2, we have now established that the family of Fisher

distributions with fixed concentration parameter K is a standard

transformation model. o

Theorem 8.1. Main theorem on transformation models. Let J be a

standard transformation model. Then, in the notation of definition 8.1

and under the topological regularity conditions specified after this

theorem the following conclusions hold:

(i)

90

~ is dominated by ~, i.e. ~« ~ and if gP

then

-1 gp(x) = peg x)

d(gP)/~

(8.28)

(ii) The maximal invariant statistic u has a distribution which

does not depend on g, i.e. u(gP) = uP for every g E G.

Furthermore, uP« u and the density of uP with respect

to u is

p(u) = f p(r,u)dp(r) <u> • (8.29)

(iii) The conditional distribution of r given u and under gP

has density with respect to p of the form

gp(rlu) ~ -1 c(u)p(g r,u) <p> (8.30)

where c(u) -1 p(u) .

(iv) If p(s,u,v) does not, in fact, depend on v then v is

distribution constant (i.e. its distribution does not depend

on g) and is independent of (s,u). Furthermore, the con­

ditional distribution of s given (u,v) and under gP

has density with respect to 0 of the form

gp(slu,v) = gp(slu) = c(u)p(g-l s ,u) <0> (8.31)

where p(s,u) denotes the density of (s,u) under P with

respect to 0 ~ u and where

-1 c(u) f p(s,u)do(s) .

<J

(v) If, in addition to the assumption in (iv), p(s,u,v) is of

the form PO(u)P1(s,w) for some nonnegative functions PO

and and where w is some invariant statistic then

(a) (s,w) is sufficient

(b) sand (u,v) are conditionally independent given w

91

given w, or (c) the conditional distribution of s

equivalently given u, and under

respect to a of the form

gP has density with

gp(s Iw)

where

-1 c(w)

-1 c(w)P1(g s,w)

S P1(s,W)da(S) . ':f

<a> (8.32)

[]

Remark. It may be shown that if one of the following two conditions

is fulfilled

(a) G has a left factorization with respect to K of the form

G HK

where H is a subgroup of G (cf. formula 2.2»

(b) K is a normal subgroup of G (cf. example 4.4»

then p(s,u,v) does not depend on v, i.e. condition (iv) of theorem

8.1 is satisfied.

Proof of theorem 8.1

For any measurable set A such that

due to the invariance of ~, that

(gP) (A) p(g-lA)

S p(x)~(x) -lA g

S p(g-lX)~(x) A

which proves (i).

-1 g A is measurable one has,

According to (c) in definition 8.1 one has for any ~-integrable

function f that

S f(x)~(x) !I

J J f(r,u)dp(r)du(u) . ~ ~

(8.33)

92

Using (i) and the G-invariance of the measure p and of the statistic

u we find for all measurable subsets U of ~ and for all g € G,

that

u(gP) (U) (gP) (u -lU ) -1 J 1u (U(X))P(g x)~(dx)

!I

J J U !'Ii

J J U !'Ii

-1 p(g r,u)dp(r)du(u)

p(r,u)dp(r)du(u)

and the proof of (ii) is complete.

To prove (iii), note that (i) and (8.33) imply that the distribution

of (r,u) under gP has density

-1 gp(r,u) = p(g r,u)

with respect to the measure p ® u. using (ii) the proof of (iii) is

easily completed.

Since gp(s,u,v) denotes the density under gP of (s,u,v) with

respect to the measure a ® u ® ~ one has

J J J gp(s,u,v)d~(v)du(u)da(s) 1. ';I ~ "t

Consequently, the condition that gp(s,u,v) does not depend on v

implies that

that ~("t) = 1

~("t) must be finite. Normalizing the measure ~

it follows that the densities of v and (s,u)

gP with respect to ~ and a ® u are, respectively,

gp(v) 1

and

gp(s,u) gp(s,u,vo) ,

where is an arbitrary element of "t. This implies that v

such

under

is

distribution constant and independent of (s,u). Furthermore, since

gp(s,u) -1 p(g s,u) ,

93

the proof of (iv) may be completed by an argument similar to that in

the proof of (iii).

Under the conditions in (v) one has that

gp(s,u,v) gp(s,u) -1

peg s,u) -1

PO(u)P1(g s,w) (8.34)

which implies that (s,w) is sufficient. Furthermore, it follows from

(8.34) that

-1 gp(u,v) = Po(u) f P1(g s,w(u»da(s)

<J

and we obtain that the conditional distribution of s given (u,v)

and under gP has derivative with respect to a of the form

-1 P1(g s,w(u»/f P1(s,w(u»da(s) .

<J

Since this expression depends on (u,v) only through w(u) the re-

maining conclusions in (v) are easily proved. 0

Example 8.1. Location-scale model (continued). From (8.11) and

theorem 8.1, or more directly from (8.3) and (8.10), it follows that

-1 n x·-f gp(x) = peg x) (s(x)/a)n IT f(-~-)

i=l a

for x € ~ and

g [~ ~] € G ,

or equivalently that

-1 n su.+x-f gp(r,u) peg r,u) (s/a)n IT f( ~ )

i=l a

- -1 where r = (s,x) and where u and g r are given by (8.12) and

(8.13), respectively. using (ii) in theorem 8.1 and (8.12) one obtains

that

p(u)

94

J~ J~ n - n-2 -_~ 0 IT f(sui+x)s dsdx

i=l

and from (iii) in theorem 8.1 that

gp(rlu) -1 n p(u) (s/a) n IT

i=l

su.+x-f f (1 )

a

<u>

<p> • o

Example 8.2. Additive effects model (continued). It is straight­

forward to check that all of theorem 8.1 applies to the additive ef­

fects model. In particular, conclusion (v) holds with w degenerate

and combining this with (8.16) and (8.19) one sees that the distribu-..............

tion of s = (a,~,~,!) has density with respect to Lebesgue measure

given by the right hand side of (8.16) multiplied by lil-(k+I+P)/2.

In view of the way in which this density is factorized, the (stochas-... .... ... ...

tic) independence of the four variates a,~,~ and ! as well as their

distributional laws can be read off directly from this result. In par­

ticular, the form of the density of !, i.e. the Wishart distribution,

is immediately seen to be

n -1' --tr{! !}

pC!:!) = cl!I-(n-k-I+1)/2Iil (n-k-I-p)/2 e 2

for some constant c independent of ! and !. A further, special

argument is required to show that

c = P I IT r (2"(f+1-v» }.

v=l

Note also that the independence of s = (a,~,~,!) and of the like­

lihood ratio statistic w for testing the assumptions of additivity

follows simply from (v) (b) of theorem 8.1, according to which s is

independent of the maximal invariant statistic u. In fact,

w = n

where I -1 - - * ! = n ! (x.. -x .. +) (x.. -x.. ) , l]V 1] l]V 1]+

from which it is seen that w

is invariant, i.e. w is a function of u only. o

95

Example 8.3. von Mises-Fisher model (continued). setting f * U(O,O,l) it follows from (8.7) that

Up(x) (8.35)

Furthermore, letting w(x) = IIx+1I formula (8.24) implies that w(x)

lIu+lI. Finally, using (8.27) we may rewrite (8.35) as

Up(s,u,v) ( () ) n Kwf· s a 3 K e .

From (v) in theorem 8.1 it follows that

<a>

i.e. the conditional distribution of s = x+/llx+1I given w = IIx+1I is

the Fisher distribution with mean direction f and concentration para­

meter Kllx+lI. Observe that this conditional model is a standard trans-

formation model with K = K. o

Example 8.4. Elliptical models (continued). We now turn to some

inferential aspects for the elliptical distributions.

Instead of considering the action of GA(p) on mP x PD(p), it is

convenient to wOrk with a free action. This is obtained by considering

the subgroup TA+(p) = T+(p) x mP, which acts transitively and freely

on mP x PD(p). Furthermore, the action of on gives rise

to the same families of elliptical distributions as is obtained by the

action of GA(p).

In the following we consider the repeated sampling situation for the

elliptical family corresponding to h, i.e. the model

where

dP (x) dx

96

Our sample space is now (mP)n and we will represent an observation

in this space by the p x n matrix

where xi € mP are column vectors. The action of TA+(p) takes the

form

* (T,E): X ~ TX + Ee

where * e (1,1, ••• ,1) € IRn. In order to apply theorem 8.1 we first

seek for an orbital decomposition.

Let

- 1 n x }; xi n i=l

* Xc X - xe

1 * 1 n * S Ii Xc Xc }; (x.-x) (x.-x) n i=l 1 1

and

~ = (X € gl(p,n) Idet(S(X» > O}

Supposing n > p we have that the complement of ~ is a closed in­

variant subset of gl(p,n) of Lebesgue measure zero and we restrict

our sample space to ~.

If X € ~ let

S (X) T(X)T(X) *

be the Cholesky decomposition of SeX) and let

where

97

(X € gl(p,n) Ix(x) 0, S (X)

Then the mapping

defines an orbital decomposition, i.e.

- * X = T(X)u(X) + x(X)e = (T(X) ,x(X» (u(X»

If dX is the restriction to ~ of Lebesgue measure on gl(p,n),

then the measure

~(X) = IT(X) I-n dX

is invariant under the action of TA+(p) on ~. This means, according

to theorem 5.4, that

(T,x,u) (J.L)

where a is the left invariant measure on TA+(p) and K is some

measure on ~. The next step is to characterize K.

For that purpose we introduce a supplementary action. Let

H = (V € SO(n) IVe = e}

Then H acts on ~ by

* V: X -+ XV

and we have that

and

- * x(XV) x(X)

* S (XV) S (X)

* u(XV ) * u(X)V

98

Furthermore,

transitively on

ant measure on

~ is invariant under the action of

•. Applying theorem 5.5 shows that

~ under the action of H. In fact

• = SO(n-1)/SO(n-1-p) = St(p,n-1) ,

i.e. ~ can be considered as a stiefel manifold.

Applying theorem 8.1 with

p(X) en

~(X) c4t

n 11

i=l

H,

K

and H acts

is the invari-

it is now a fairly trivial matter to see that the density of (T,x)

conditionally on u = {U1 , ... ,Un } and with respect to the left invari-

ant measure on TA+(p) is given by

p(T,x;f,};lu) =

n _ * -1 _ 11 h«Tui+x-f)}; (TUi+X-f»

i=l

and that u has density

p (u) n _ 2 _ 11 h(IITUi+xll )da(T,x)

i=l

with respect to the invariant measure on ~.

(8.36)

In particular, if h(s) = (2v)-p/2 exp(-~ s) we obtain the multi-

variate normal model, where (8.36) depends on

that (T,x) is independent of u and that u

buted on the Stiefel manifold ~.

(T,x) only, showing

is uniformly distri­

[]

Example 8.5. Elliptical MANOVA models. Example 8.4 is easily gener­

alized to cover a MANOVA model for an elliptical family. Therefore, we

only give the details necessary for completing the discussion of these

models.

Let

JI trW € gl(p,n) Ir € gl(p,q)} ,

99

where W is a known q x n (design) matrix of rank r. Suppose n-r

~ p, and let Q denote the n x n matrix of the orthogonal

projection onto the subspace of ffin spanned by the rows of W, i.e.

MQ M, M € .M ,

and

NOw, G T+(p) x.M acts on gl(p,n) by

(T,M): X ~ TX + M .

Letting

M(X) XQ ,

* (X € gl(p,n) Idet(X(In-Q)X ) > O} ,

S (X) T(X)T(X) * T(X) € T+(p) , X € ~ ,

and

(X € gl (p,n) IM(X) 0, S (X)

it follows that

X ~ «T(X) ,M(X», u(X»

is an orbital decomposition. Next, the group

H = (V € O(n) IQv* = Q}

acts on !t * by V: X -+ XV

100

and transitively on ~ * by u -+ uV .

Furthermore, (T,M) is invariant whereas u is equivariant under H.

Finally

~(X) = IT(X) I-n dX

is invariant under the action of G x H.

It is now straightforward to continue in the same way as we did in

example 8.4. It may be noted that example 8.2 is also covered by the

present model. o

Example 8.6. Conical surface model. In the following we consider a

class of distributions on IRq,{O} = !to {["~"] € IRq+llx € IRq, x 'I- o}

which has recently been characterized by invariance, cf. Letac (1988).

This class of distributions is a subclass of the mUltivariate general­

ized hyperbolic distributions, see Barndorff-Nielsen (1977), Bl~sild

(1981), and Bl~sild and Jensen (1981) p. 65. In the present context we

will refer to this class of distributions as the conical surface dis­

tributions in IR q+1 •

According to example 6.7 we have that IR: x sol (l,q) acts transi-

tively on !to by

and that

-1 ~(x) = IIxll dx

is relatively invariant with multiplier ~l(a,A)

Consider the probability measure

where (1,0, ... ,0) € IRq+l,

J IIxll-1 e- lIxll dx

IRq,{O}

t(x) * (lIxll, x )

q-l a .

and

Here

101

J dv J~ r-1 e-r r q - 1 dr sq-1

aIr (q-1) . q-

is the surface area of the unit ball

Transforming P by (a,A) yields

d«~A)P)(X) = a-q+1 c q (-1(A-1)* t()) exp -a eO· x

If -1 -1 * 9 = a (A ) eO it follows from example 6.8 that

9

and we clearly have that 9*9

d( (~A) P) (x) = C q (9*9) (q-1)/2 exp(-9.t(x))

where -1 -1 * 9 = a (A ) eO'

9

-2 a

traverses

i.e.

Subsequently, we consider the situation where we have a sample

x 1 , ... ,xn ' n ~ 2, from this class of distributions.

Letting (a,A)P = P9 and ~ = (x1 , ... ,xn ) we have that

n ! t(xi ), and it is easily seen that

i=l

As pointed out for the elliptical distributions, it is convenient to

work with a free and transitive action on 9. For that purpose we

* restrict ourselves to consider the action of G = ffi+ x P(q), where

P(q) is defined in example 3.6. Furthermore, we restrict our sample

space to ~ = {~It+(~)*t+(~) > OJ. This leaves out a closed invariant

subset of Lebesque measure zero, and G acts freely on ~.

102

Next, we want to describe the distribution of the sufficient statis­

tic t+ and the orbit projection v, i.e. we want to transform p:n

by the mapping

In order to apply theorem 8.1 we have to find an invariant measure

on ~n ~. The measure ~ is relatively invariant with multiplier

J(n (a ,A) n(q-1) a , i.e. we need a modulator which is invariant under

(l,A), A € P(q). Now t+(~)*t+(~) is the maximal invariant under the

action of P(q) on ~ and it is easily seen that

We may conclude that is a modulator with J(n

as associated multiplier. Hence

is invariant and

Finally, we need to describe the invariant measure T on ~, which

by example 6.8 is given by

dT(t) = (t*t)-(q+1)/2 dt

Application of theorem 8.1 now yields that v and t+ are independent

and that the distribution of t+ is given by

c (S*S)n(q-1)/2 (t*t)n(q-1)/2-(q+1)/2 n,q

-s·t e

(8.37)

where c n,q

103

is a normalizing constant, which is independent of a.

Setting a = (1,0, ... ,0)* € mq+1 in (8.37) a direct calculation

shows that

c n,q

-qj2 1 T f(2(n(q-I)+1))

f (~(n-I) (q-I))f (n(q-I)) []

Example 8.7. Conical model. In continuation of the preceeding ex­

ample, we focus on a family ~ of distributions as described in

(8.37), Le. !f1 = (Pala€9} and

dP '\ (a*a)A.(t*tl" -a·t _aCt) e (8.38)

dsL ,q

Here t € ~ and

~ (t € * m+ x mqlt*t > O}

9 {a € * m+ x mqla*a > O} ,

and

dsL (t) (t*t)-(q+I)/2 1 (t*t) dt (0, (X»)

is invariant under the action of

well-defined for all A. > (q-1)/2

(8.38) is given by

'\,q f(A.-~(q-I))f(2A.)

* G = m+ x P(q) on ~. The family is

and the norming constant '\,q in

We proceed to investigate a submodel of ~ which is invariant under

P(q). Any such model corresponds to a fixed value of the maximal in­

variant, when we consider the action of P(q) on 9. A maximal in­

variant is given by a*a, and consequently (a*a)1/2 = KO corresponds

to the submodel

dP Ko,m dJL (t)

104

where m € Hq . When we are making inference for a, we should condi­

tion on the maximal invariant, and hence ancillary, statistic ret) = (t*t)1/2. (In the terminology of theorem 8.1 the maximal invariant

statistic is denoted by u. Here we use the letter r for the maximal

invariant statistic (t*t) 1/2 since this statistic is in fact the

(hyperbolic) resultant length of the vector t.) If t ~ (r(t),s(t»,

where set) r(t)-lt € Hq , we have by example 6.8 that

(r,s)(~) = p ® 0

where 0 is invariant on Hq . It follows that the distribution of s

conditionally on r has density with respect to 0 given by

Here c (.) is a normalizing constant given by q

C (K) q

(K/211") (q-1)/2

2K(q_1)/2(K) (8.39)

where K(q-1)/2 denotes the modified Bessel function of the third kind

and with index (q-1)/2.

This class of distributions has been considered for observations on

Hq , i.e.

dP d~ ,m) (s) C (K)

q

and is known as the hyperboloid model, see Jensen (1981). However,

Jensen writes the density in terms of the * product in the following

way

dP (K,T}) (s) = c (K) do q

-K11*S e ,

and refers to the parameters 11 = 11 m E Hq ,q and K E IR+

(8.40)

as the direc-

tion parameter and the concentration parameter, respectively.

105

Note that the family is a transformation model only when K is

fixed. In the case of a sample s1, ... ,sn this means that it is more

difficult to derive the distribution of the minimal sufficient statis-n

tic 5 = 1 }; n i=1

since we must also characterize the quotient

measure corresponding to the maximal invariant 5*5. See the continua-

tion of this example at the end of the section. 0

Example 8.8. von Mises-Fisher matrix model. In the following we

give a brief introduction to the socalled von Mises-Fisher matrix dis­

tributions, see Downs (1972) and Jupp and Mardia (1979). These distri­

butions constitute a model for observations on the stiefel manifold

* St(p,n) = (XEgl(p,n) Ixx =Ip }' i.e. observations of p mutuallyortho-

gonal directions in ffin.

If dX

st(p,n)

given by

denotes the O(p) x O(n)-invariant probability measure on

- as described in example 4.6 - the family in question is

~ = (PMIM E gl(p,n)} with

* c(M) exp(tr(MX »

where tr indicates the trace and C(M)-1 = oF1 (n/2,MM*/4) is a

hypergeometric function of matrix argument, cf. James (1964).

Since the action of O(p) x O(n) on St(p,n) is given by

* (U, V): X -+ UXV

it is simple to see that

d( (U, V) PM) dX (X) * * a(M)exp(tr(UMV X »

i.e. the induced action on the parameter space ~

by

* (U , V): M -+ UMV

gl(p,n) is given

This action has been considered in example 2.7, where A1 (M) ~ ... ~

2 P {Ai (M) } i=1 being the eigenvalues of

ized as a maximal invariant.

* MM , are character-

106

If is fixed, it follows that ~A o

~A = (M € gl{p,n) IMM* o

formation model and that

* c(Ao)eXp(tr(MX »

where the norming constant

diag(A~l, ... ,A~p'O, ... ,O) € gl(n,n), depends on AO only.

is a trans-

As noted in example 2.7 there are multiple types of orbits, i.e.

essentially different models, depending on the set of equalities in the

relation A01 ~ •.. ~ AOp ~ o.

We do not proceed further, but note that the von Mises-Fisher model

on the sphere corresponds to p = 1, and that the complications exhi­

bited in example 8.3 do not at all simplify in this more general set-

ting. 0

We finally turn to a brief discussion of composite transformation

models, as defined at the beginning of this section. such a model con­

sists of a class of probability measures ~ which may be partitioned

as ~ {~A: A € A} for some index set A with each subclass ~A

being a transformation model relative to a group G acting on the

sample space ~, the group G and its action on ~ being the same

for all A. Thus ~A is of the form ~A = {gPA: g€G}. The variate A

is called the index parameter of the model.

Since the group G and its action on ~ are assumed to be indepen­

dent of A € A it seems natural to require that the same holds for the

quantities in definition 8.1 related to the action of G on ~. Con­

sequently, we assume that the quantities ~, u, v, K, r, p and ~

do not depend on A € A. The remaining quantities in definition 8.1

may depend on A and in that case we write PA' etc.

We then say that ~ is a standard composite transformation model pro­

vided the only quantities which do, in fact, depend on A are PA and

p(·;g,A), where

107

and we speak of ~ as a balanced standard composite transformation

model if, in addition, K K.

Let ~ = (~X: X€A) be a standard composite transformation model and

suppose that the conclusions of theorem 8.1 apply to each of the trans­

formation models ~X. In particular, one then has that

-1 p(x:g,X) = peg x:e,X) (8.41)

and, furthermore, that the marginal distribution of u has density

with respect to u of the form

p(u:X) = J p(r,u:e,X)dp(r) <u> • (8.42)

In many contexts of statistical inference interests centers on like­

lihood functions. In the present setting the primary likelihood func­

tion is

L(g,X:x) p(x:g,X) (8.43)

considered as a function of (g,X) for fixed x. Factors in L de­

pending on x alone are considered irrelevant. Note, however, that

according to assumptions one has that Gp = K. Consequently, the X

density function p(x:g,X) depends on g only through gK or equiva­

lently only through s, where s denotes the point in ~, considered

as the parameter space, corresponding to gK (cf. comment (B) to defi­

nition 8.1). Thus the primary likelihood function (8.43) may be re­

written as

L(S,X:X) p(x:s,X) (8.44)

Similarly,

L(X:u) = p(u:X) , (8.45)

with p(u:X) given by (8.42), is a marginal likelihood function for X

based on observation of the maximal invariant statistic u alone,

factors depending on u (or even x) alone being again considered

irrelevant.

We shall now show that the formula (8.45) can be transformed into

another expression for the marginal likelihood L(X:u) in terms of an

108

integral of the primary likelihood L(S,A;X), but an integral with

respect to s (or, equivalently, a part of g) rather than r (or,

equivalently, a part of x). The disregarding of parameter free fac­

tors is essential in this connection and the resulting formula (8.46)

is often simpler to apply than (8.45). Specifically we have

Theorem 8.2. Let 1 = {1A: A € A} be a standard composite trans­

formation model with index parameter A. Suppose that for every A E A

the conclusions of theorem 8.1 apply to ~A and that, in addition, the

isotropic group K is compact. Then the marginal likelihood L(A;U)

based on the maximal invariant statistic u may be calculated as

~ -1 ~ ~ L(A;U) = f L(S,A;X)A G (s)da(s) . (8.46)

In (8.46) a denotes the invariant measure on ~ and the modular

function AG of the group G is considered as a function of s, or

equivalently as a function of the left coset gK. This function is

well-defined because of the compactness of K which implies that

AG(k) = 1 for every k € K. o

Proof. The basic formula used in the proof given here of theorem

8.2 is formula (5.25) stating that

where H is a closed subgroup of G and where a G/ H denotes the

invariant measure relative to the natural action of G on G/H.

Since K is compact it follows from the assumptions that K is

compact and consequently that aK(K) < 00. From (8.45) and (8.42),

using (8.47) with H = K after a transformation, it follows that

L(A;U) P(U;A)

f p(r,u;e,A)dp(r) ~

Disregarding the factor

(4.4) and (4.3) we find that

L(A;U) J P(gK,u;e,A)daG(g) G

~ -1 J p(K,u;g ,A)daG(g) G

109

and using, respectively, (8.41),

~ -1 J p(K,U;g,A)A G (g)daG(g) G

NOw, let go € G be such that x = gou. From the invariance of the

measure a G and from (8.41) we obtain that

~ -1 -1-1 L(U;A) = J p(K,u;go g,A)A G (go g)daG(g)

G

where in the last equality we have used the fact that the quantities

P(goK,U;g;A) and AG(g) depend on 9 only through gK. This fact in

conjunction with (8.47) with H = K implies that

110

Disregarding the factor AG(go)aK(K) and transforming we find that

L(u;X)

~ -1 ~ ~ f p(X;s,X)A G (s)da(s) ';/

~ -1 ~ ~ f L(S,X;X)A G (s)da(s) , ';/

as was to be proved. o

The importance of theorems 8.1 and 8.2 is intimately connected with

two basic principles of statistical inference, those of conditionality

and of marginalization. According to the first, inference on g, or on

that part of g on which gp genuinely depends i.e. gK, should be

performed conditionally on a suitable ancillary statistic. Under the

assumptions of theorem 8.1, (u,v) constitutes such a statistic and

the conditional model given (u,v) reduces to that determined by the

model function gp(slw), for which the formula (8.32) is available.

The principle of marginalization implies that under a composite trans­

formation model the proper basis for inference on the index parameter

X is the marginal distribution of the maximal invariant statistic u.

The density function for this distribution is given by (8.42), and

while that formula is often rather intractable the derived expression

(8.46) for the marginal likelihood is more manageable.

We close this section by showing an application of formula (8.46).

Example 8.7. Conical model (continued). As mentioned above, the

class of hyperboloid distributions p ~ given by (8.39) and (8.40) (K, s) ,

and with direction parameter s € Hq and concentration parameter

K € ffi+, is a standard composite transformation model with K as index

parameter. The likelihood function corresponding to a sample ~

(sl, ... ,sn) of size n ~ 2 is

dp@n ~ (K,S)(S) @n -

da (8.48)

111

where s = + n I

i=1 si. Expressing the mimimal sufficient statistic s+

in terms of the maximal invariant statistic u = (s *s ) 1/2 + + -maximum likelihood estimator of s one obtains

and the

Inserting this in (8.48) the primary likelihood function takes the form

(8.49)

Since a is the SOf(1,q)-invariant measure on the parameter space ~ Hq it follows immediately from (8.39), (8.40), and (8.49) that the

marginal likelihood (8.46) for K is

L(K ;u)

where c (.) q

(8.50)

is given by (8.39).

Formula (8.50) may be compared with the actual distribution of u.

By formula (8.42) this distribution has density function with respect

to Lebesgue measure of the form

with L(K;U) as in (8.50). Rukhin (1974) has shown that

co

h 2 (U) = wn+1 ~e(in+1 f (Hg1 ) (X)}nJo(UX)XdX) , o

(8.51)

(8.52)

where is a Hankel function and J o is a Bessel function, - a

rather redoubtable expression. In contrast, for q = 2 we have simply

(2w)n-1 n-2 (n-2)! u(u-n) ,

cf. Jensen (1981). For q > 3 the form of is not known.

If we apply the asymptotic relation

112

~ -1/2-x K (x) ~ vV/2 x e v

x ~ 00

to (8.50) we find that

Suppose we adopt the right hand side expression as an approximation for

L(K;U). This expression is recognised as the likelihood function of a

gamma distribution and this suggests that 2K(u-n) is approximately

distributed as ~2«n-1)q), i.e.

(8.53)

A more detailed calculation shows that this indeed the case, see Jensen

(1981). As mentioned above, the exact distribution of u is known only

for q = 1 and 2. In the latter case (8.53) is, in fact, exact and

for q = 1 it represents a considerable simplification over the exact

result, cf. (8.50)-(8.52). o

Bibliographical notes

Some key references to the statistical literature on transformation

models are Barnard (1963), Fraser (1979), Barndorff-Nielsen, Bl~sild,

Jensen and J0rgensen (1982) and Barndorff-Nielsen (1983, 1988).

Theorem 8.1 is an extended version of theorem 3.1 in Barndorff-Nie1-

sen, Bl~sild, Jensen and J0rgensen (1982).

In the balanced case, i.e. for K = K, the results of theorems 8.1

and 8.2 are stated in Barndorff-Nielsen (1988), chapter 2.

Further results and exercises

~ Let G be a group acting on ~ and assume there exists a mapping

z: ~ -+ G

which is equivariant , i.e. z(gx)

G. Show that the mapping

gz(x) for all x € ~ and all g €

u: ~ -+ ~

x -+ (z(X»-lx

is maximal invariant.

h i)

Let M be a linear subspace of IR n

* Show that G = IR+ x M acts on

of dimension

IR n by

G x IR n -+ IR n

([a,JL] IX) -+ aX+JL

[Section 2]

m < n.

ii) Let p denote the orthogonal projection on M with respect to

the usual inner product on IR n and define

s(x) = IIx-p(x)lI.

Show that G acts freely on

Show that the action of G on

is transitive but not free and determine the isotropy group at

o.

114

iii) Let

{x€3: I s(x) 1, p(x)=O}

and define

u: er ~ 'tI

x ~ S(X)-l(X-p(X»

Show that

x ~ ([s(x),p(x»), u(x»

is an orbital decomposition.

[Section 2]

~ Let PD(n) denote the set of positive definite n x n matrices

and T+(n) the group of upper triangular matrices with positive

diagonal elements.

i) Show that T+(n) acts transitively and freely on PD(n) by

T+(n) x PD(n) ~ PD(n)

(T,~) * ~ T~T

(Consider the Cholesky decomposition of ~).

ii) Let P(k) = (m:)k denote the multiplicative group of positive

vectors, i.e.

X,TJ € P(k) .

Show that G P(k) x T+(n) acts freely on PD(n)k+1 by

G x PD(n)k+1

«T,X), (SO,Sl, ... ,Sk»

115

iii) * Let So = TOTO' TO € T+(n) be the Cholesky decomposition of

So' and define

i 1, ... ,k

Show that

is an orbital decomposition.

[section 2]

~ Let SL(2) be the special linear group {A€GL(2) Idet(A)=l} and

consider the action of SL(2) on gl(2) - the vectorspace of 2 x 2

matrices - given by

Let

and

Show that

SL(2)xgl(2) ~ gl(2)

* (A,B) ~ ABA

* {B€gl(2): B=B }

* {B€gl(2): B=-B }.

+ -gl (2) = gl (2)61g1 (2) ,

i.e. gl(2) is the direct sum of gl+(2) and gl-(2).

Show that gl+(2) is invariant under the action of SL(2), i.e.

if A € SL(2), B € gl+(2) .

116

similarly, show that gl-(2) is invariant and that the action of

SL(2) on gl (2) is trivial.

Show that gl+(2) has 4 orbit types corresponding to the following

conditions on B.

i)

ii)

iii)

iv)

B o .

Show that GB o and that GB SL (2) •

det(B) = 0, B ~ O.

Let ~ € {±1} denote the sign of the non-zero eigenvalue of B.

Show that

u(B) = [~ g]

is an orbit representative and that

~] la € ~, 0 € {±1}}

det (B) A > 0 •

Let ~ € {±1} denote the sign of the eigenvalues of B. Show

that

u(B) = ~~ [~ ~]

is an orbit representative and that GU(B)

Show that

det(B) A < 0 •

u(B) = J=X [01 0] -1

is an orbit representative and that GU(B)

So (2) •

SO(l,l).

[Section 2]

117

~ Let (G,~) be a transformation group.

Show that the following conditions are equivalent to (G,~) being

standard respectively to properness of the action. 01)

Suppose that {(Yn,Xn)}n=l ~ v and (Yn'xn ) ~ (Yo,xo ) € ~x~. n~

Standard: Then there exists a sequence 01)

{gn}n=l ~ G such that

01)

Properness: For every sequence {gn}n=l so that Yn 01)

have that {gn}n=l has a convergent subsequence.

[Section 2]

~ Let H be a closed subgroup of G and consider the action

Show that -r is proper.

-r: H x G ~ G (h,g) ~ hg

[Section 2]

we

~ Show that the action of G on ~ is proper if G is compact.

[Section 2]

~ Let H be a closed subgroup of G and consider the action

-r: H x g ~ G (h,g) ~ hgh-1

Show that properness of -r is equivalent to H being compact.

[section 2]

~ Let (G,~) be a transformation group and H a closed subgroup of

G.

Show that if (G,~) is standard respectively G is acting proper­

ly then - with the inherited action of H - (H,~) is standard respec-

118

tive1y H is acting properly.

[section 2]

10. If A E gl(n) then there exists an upper triangular matrix T -1 and an invertible matrix S such that A = S TS (the Jordan normal

form). Note that Sand T have some complex entries in case A has

some complex conjugate eigenvalues.

i) Show by referring to (3.6) that

det(exp(As» = exp(tr(A)s) , S E IR .

ii) Show that the Lie algebra of SL(n) = (SEGL(n) Idet(S)=l} is

given by

sl(n) (AEg1(n) Itr(A)=O} .

[section 3]

11. Reconsider exercise 4 and the 3 orbit types under the action of

SL(2) on gl+(2)\[0]. The Lie algebra sl(2) of SL(2) is given by

exercise 10.

i) Consider the set

~1 (BEg1+(2) Idet(B»O}

with constant orbit type, and determine the Lie algebra ~ of

Show that

s1(2) st (2) ® 'j{

where

st(2) = {[~ _~] la,b E IR} .

Show that st(2) is the Lie algebra of the group

ST(2) = {[~

119

Finally, show that ST(2) acts freely on ~1' i.e. the equa­

tion

B * Tu(B)T T E ST(2)

uniquely determines an orbital decomposition.

ii) Consider the following sets, each having their own orbit type:

~2 {BEgl+(2) Idet(B) = 0, B#O}

~3 {BEgl+(2) Idet(B)<O} .

Identify in both cases the Lie algebra ~ of the isotropic

group GU(B) and show that

sl(2) so(2) ~ :1/ ~ ~

where

This leads to considering

SO(2) = {V(S) = [c~SS -sinS] Is E [0 2W)} s~nS cosS '

and

Show that in both cases the system

* B V(S)O(a)u(B)O(a)V(S) S E [O,w), a > 0

has a unique solution, i.e. determines an orbital decomposition.

The similarity between these two decompositions indicates that

the concept 'constant orbit type' might be relaxed by only re­

quiring isotropy groups to be isomorphic, as is the case in this

example.

[Section 3]

120

12. Let G be the subgroup of GL+(3) determined by

G

Determine /3 G and

13. Let G be the subgroup

{ [~ 0

G Y 0

of GL (3) . Find a G , /3 G and

Let H be the subgroup of

H { [g

o x o

~] :

AG·

G

0

Y 0

~]: x > 0, y,z E rn} .

[sections 4 and 6]

x,y,z E rn, xy "F o}

determined by

~] : y "F o} .

Show that there exists no G-invariant measure on G/H.

[Sections 4 and 6]

14. Let G be the subgroup of GL+ (3) given by

{[~ 0 log X]

> 0, y,z E rn} . G x z : x 0 1

Find a G , /3 G and AG·

Let H be the subgroup of G determined by

H = {[~ 0 ~]: y E rn} . 1 0

Show that H is unimodular and that there exists an invariant measure

under the natural action of G on G/H. Determine this measure.

[Sections 4 and 6]

121

Consider the action of on

GA+(n) x ~n ~ ~n

([A,f] ,x) ~ Ax+f •

given by

Let ~ be the measure corresponding to the n-dimensional normal

distribution with mean 0 and variance In' i.e.

~ (dx)

where ~n denotes Lebesgue measure on ~n and where

Show that ~

plier l([A,f],x)

is a quasi-invariant measure. Find the quasi-multi­

and determine the function p in formula (4.8).

[Section 4]

16. Let Hand K be closed subgroups of G such that the action

H x K: G ~ G

(h,k): g ~ hgk-1

is proper and transitive. Let a G be a left Haar measure on G.

Show that - under the action of H x K - the measure is rela-

tively invariant with multiplier l(h,k)

NOW, let ~ € meG) be determined by

~ (f)

where and are left Haar measures on H and K, respective-

lye

Show that ~ is equal to a G, up to a constant, and summarize the

result in the formula

122

[Section 4]

17. Let ~ be a d-dimensional manifold and let (V,~) be a local

parametrization. Recall that for x € ~(V) the corresponding coordi­

nate frames E1x , ... ,Edx of T!x' the tangent space at x, are given

by

-E...... (fo~) (v) , av1

where f is a smooth function defined in a neighbourhood of x = ~(v). Let E1X , ... ,Edx be the frames corresponding to an alternative

local parametrization (Z,C). Show that using Einsteins summation

convention one has

where v and

Let ~ be a Riemannian manifold with metric ~, i.e. for every x

€ ~ the value ~x of ~ at x is a positive definite, bilinear form

on T!x' which in the local parametrization (V,~) is represented by

Show that

where refers to the representation of ~ with respect to the

local parametrization (Z,C).

Finally show that the geometric measure a on ~ given by formula

(5.5) is a geometrical quantity, i.e. it does not depend on the local

parametrization.

[section 5]

123

18. Consider the cone

as a Riemannian submanifold of i.e. the metric on is the

restriction to ~ of the usual Riemannian (or Euclidean) metric on

m3 • Let u be the mapping

u: ~ -+ m+ x -+ ( * ) 1/2 x x ,

where

For every u € m+ find the geometric measure on

~u {x€~lu(x)=u}.

Consider m+ as a Riemannian submanifold of m and find the de­

composition of Lebesgue measure on ~ determined by formula (5.7).

[Section 5]

19. Let ~ be a d-dimensional Riemannian manifold with metric ~. A

local parametrization (Z,C) is called orthonormal at x € ~ (with

respect to ~) if for x = C(z) one has

{jab '

where {j denotes Kroneckers delta function and where the d x d ma-

trix is the representation of

system z.

~ x in the coordinate

Let (V,~) be an arbitrary local parametrization and let q(v) be

the corresponding representation of ~. For a fixed x € ~(V) let v

= ~-1(x) and let q1/2(v) be a square root of q(v), i.e. q1/2(V)

is a d x d matrix satisfying q(v) = q1/2(V)q1/2(v)*. Show, using

exercise 17, that if z: V -+ Rd is a one-to-one mapping such that

124

* az (v) av q1/2 (v) ,

then one obtains an orthonormal local parametrization (Z,C) at x by

letting

Z z (V)

and

C = .;ov

where v = z -1 the inverse mapping of z. Now let, in addition, u be a differentiable mapping from ~ into

some m-dimensional Riemannian manifold ~ with m < d and metric ~.

Let (V,~) be a local parametrization at u(x) such that ~(V) = u(';(V» and let (Z,r) be a i-orthonormal parametrization at u(x)

constructed as above. Thus, expressed in terms of the local coordinate

systems v and v, the mapping u becomes

and expressed in the coordinates z and z u is given by

With this notation it follows from formula (5.7) and the remarks imme­

diately after the formula, that the differential OU and the modulat­

ing factor evaluated at x € ~ are given by, respectively,

OU

and

louou*I-1/ 2

* az

a" a"* -1/2 I u* ~I az az

Show that these quantities expressed in terms of the arbitrary

coordinates v and v become, respectively,

Ou

and

125

q1/2(V} aK* q-1/2(v} av

Consequently, the factor louou*I-1/ 2 in formula (5.7) may be calcu­

lated without reference to orthonormal parametrizations.

[section 5]

20. Let ~ be a family of probability measures on ~ dominated by a

a-finite measure ~. Identify ~ with the set of densities ~ =

{~(-) Ip€~} and assume that ~ (or ~) is a d-dimensional manifold.

Let

w -+ p(- ;w} dP w ~ (.)

be a parametrization of ~. Show that the Fisher information i(w}

{irs(w}}, given by

defines a Riemannian metric on ~ (provided that i(w) is positive

definite} .

[Section 5]

21. Identify the set of normal distributions H = {N(f,a 2 }: fEffi, a 2>0}

with the set ffi+ x ffi = {[a,f]: a>O, fEffi} and consider the Fisher in-

formation on H. Show that the geometric measure , on H correspon­

ding to the Fisher information satisfies

, = F2 a GA (1) +

and conclude that , is invariant under the action of GA+(l} on H

given, in obvious notation, by

126

GA+(l) x if .... if

([;:T ,f] ,N(f,a2 )) .... ;:TN(f,a 2 ) + f [section 5]

22. Consider the unit hyperboloid

as a Riemannian submanifold of ffi3 and show that the geometric measure

on H2 is given by

Prove, using (8.39), (8.40) and the formula

that the density function of the hyperboloid distribution on H2 with 2 direction parameter ~ = (~O'~l'~2) € H and concentration parameter

K € ffi+ can be expressed as

Prove (or take for granted) that the corresponding Fisher informa­

tion is

-2 0 0 K

2 - (1+K)~1~2

i (K '~l'~2) 0 (1+K) (1+~2)

2 2 ~O ~O

- (1+K) ~l ~2 2

0 (1+K) (1+~1)

2 2 ~O ~O

127

Now assume that K is known. Show that the geometric measure a i

on corresponding to the Fisher information for the

unknown parameters is given by

[Sections 5 and 8]

23.

Then

Let G be a closed subgroup of GL(n), i.e. G is a Lie group.

g-l dg determines a set of left invariant one-forms on G, i.e.

-1 g dg.

These may be used to construct a left Haar measure on G, as de­

scribed on p. 72.

Show that (dg)g-l may be used for the construction of a right

Haar measure on G.

A simple example is provided by GA+(l), where

{~

determines -2 a dad,l as left invariant, whereas

}-1 {-I Jl _ a da 1 - 0

determines -1 a dad,l as right invariant.

Use this method to determine left and right Haar measures on the

groups considered in exercises 12, 13 and 14.

[Section 7]

24. Let S € PO(p), the set of p x p positive definite matrices.

Recall from p. 70-71 that if

* S T T

where T € T+(p), is the Cholesky decomposition of S then

dS " ds .. i~j 1J

128

The p-dimensional gamma function fp is given by

for A > (p-1)/2.

Show that

S IsI A-(p+1)/2 e-trs dS PO(p)

I1 P (p-1)/4 P 11 f(A-1/2(i-1))

i=1

and conclude that for ! € PO(p) one has that

Thus setting A = f/2, where f is an integer greater than or equal

to p, one obtains the density function of the p-dimensional Wishart

distribution with f degrees of freedom and variance !

1 I S I (f-p-1)/2 2Pf/2 f (f/2) 1!lf/2

P

1 -1 -2' tr! S e S € PO (p) .

Show that the maximum likelihood estimator ! on p. 94 is dis­

tributed according to a p-dimensional Wishart distribution with f = 1 n-(k+1-1) degrees of freedom and variance n!.

[Sections 7 and 8]

25. This exercise is concerned with the structure of exponential

transformation models. More detailed discussions of such models may be

found in Barndorff-Nielsen, Bl~sild, Jensen and J0rgensen (1982), Erik­

sen (1984) and Barndorff-Nielsen (1988). For notation and terminology

concerning exponential families the reader is referred to Barndorff­

Nielsen (1978).

129

Let 1 be a standard transformation model on ~ with acting group

G and assume, in addition, that 1 is an exponential model of order

d with minimal representation

-1 p(x;g) = peg x) a(9(g»b(X)e9 (g).t(X)

~ being a G-invariant measure on ~. Furthermore, assume that the

minimal sufficient statistic t is continuous.

i) using the affine relationship between respectively the canonical

parameters and the canonical statistics of two minimal represen­

tations of 1 (cf. for instance lemma 8.1 in Barndorff-Nielsen,

1978) show that there exist uniquely determined subgroups G and G of GA(d),

( [A (g) , B (g) ]: gEG}

G * -1 ~ {[A (g ) ,B(g)]: gEG}

such that:

a) t(gx) A(g)t(x)+B(g)

b) 9(g)

and

c) the mappings

G .... G

9 .... [A(g) ,B(g)]

and

G .... G

* -1 ~

9 .... [A (g ) ,B(g)]

are representations of the group G, i.e. the mappings are

homomorphisms from G into GA(d) .

130

ii) Let

iii)

6(g) a(9(e))/a(9(g))e-9 (g).B(g)

and prove that

Let E.(g)

given by

~ -1 b(gx) = 6(g)b(x)eB(g )·t(x)

In 6(g) and let M(g) be the element in GL(d+2)

M(g)

A(g)

o ~* -1 B (g )

B(g) o

1 o

E. (g) 1

Show that the mapping

9 -+ M(g)

is a group representation of G, i.e. a homomorphism from G

into GL(d+2).

iv) Let +(x) = In b(x) and let s denote the (d+2)-dimensional

statistic given by

* * s (x) (t (x),1,+(x))

Furthermore, let (g,u) be an orbital decomposition of x,

i.e. x = guo Show that

s(x) M(g)s(u)

and, in conclusion, show that there exists a constant (d+2)-di­

mensional vector c such that

p(x) = eC·M(g)S(U)

v) Let e denote the parameter space of the full exponential fami­

ly ~ containing ~, i.e.

131

Show that the group G = ([A*(g-l),B(g)] IgEG} leaves e in­

variant and conclude that if the action of G on e given by

(G,e) -+ e (g,9) -+ A*(g-1)9+B(g)

is not transitive then ~ is a composite transformation model.

[Section 8]

26. Denote by ~ the 2 x 2 matrix

{ C1 -1 C1 -C1

and consider the group of 2 x 2 matrices

G = {~:C1>o}

with the usual matrix multiplication as the rule of composition. Let­

ting (xl, .•. ,xn ), xi E m2 , denote a point in gl(2,n) we define an

action ~ of G on gl(2,n) by

Now let ~ denote the transformation model on gl(2,n) generated by

G and by the probability measure Po under which x 1 , ... ,xn are n

independent observations from a bivariate normal distribution with mean

o and variance equal to the identity matrix.

i) Determine the set ~ where the maximum likelihood estimate C1

of C1 exists uniquely.

ii) Show that G acts freely and properly on ~ (see exercise 5)

and that the complement ~c is closed and of Lebesgue measure

zero.

132

iii) Show that the Lebesgue measure on ~ is invariant under the

action of G.

iv) Find a maximal invariant statistic u (see exercise 1).

v) Find the conditional distribution of the maximum likelihood '2 2 estimate a of a given the maximal invariant statistic u.

[Section 8]

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Sankhya 48, 1-42. [14]

Subject index

action 3,14,43,46,47,48,49,54,74 free 4 induced 74

137

instances 3,4,5,5-6,7,8,9,10,22,26,38,39,49,50,51,59,65,67,75,76, 77,78,84,86,95,96,99,100,101,103,105,113,114,115,117, 118,119,121,125-126,131

left 10,32,52,55 natural 11,36,52,53,108 proper 13,36,45,49,117,118,121,131 right 10,32 transitive 3 see also group; conjugation

additive effects model 76,85-87,94-95

analysis of variance see additive effects model

ancillarity principle 82

ancillary statistic 82,110 instances 84,104

Bessel function 77,104,111

Blaschke-Petkantschin formula 59-60

boost 24,39

canonical parameter 129 statistic 129

Cauchy distribution mUltivariate 79

chart 15

Cholesky decomposition 70,96,114,115,127

commuting mapping 48

composite transformation model 75,78,106,110,131 standard 106,108

balanced 107

concentration parameter 76,104,126

conditional distribution

formulae 90-91 independence 90

conditionality principle 110

conditioning 82

cone 49,123

configuration statistic 84

conical model 103-105,110-112 surface model 100-103

conjugation 56

coordinate frame 45,122

coset left 11 right 10

138

decomposition of measures 1,42,45,46,47,48,74,123

differentiable manifold 15,66,68,72,122,123,124,125

see also manifold submanifold 54,123,126

differential 45,66,124 r-form 71

invariant 72,127 of function 69

direction parameter 104,126

disintegration 1,2,42,53 formulas 42,46,48

distribution constancy 90

Einstein summation convention 69,71,122

elliptical models 78-79,95-98 MANOVA models 98-100

equivariant mapping 4,80,113 statistic 80,84,88,89,100

exponential map 15,18,21

instance 24 model 129

representation of 129

exterior calculus 1,66

exterior product 69 and Jacobians 69 and invariant measures 72

factorization of group 15,21,27 left 11,52,72,82,91

of GL(n) 38 of SO(p) 22

of sol (1,q) 23

of SOl (p,q) 26 right 10

of GL(n) 38

of sol (p,q) 23

139

factorization of measures 1,46,48,66 of Lebesgue measure 37,66,67-68 of left invariant measure 52,121-122 of lifted measure 42,48,49,50

instances 51,52,65 of right invariant measure 52

Fisher distribution information

87-89,95 125,126,127

forms see differential

gamma distribution 112 function 128

geometric measure 45,53,54,122,123,125,126,127 on GA(n) 56 on GL(n) 56 on PD(n) 58 on sp-1 62 on T+(n) 57

G-invariant measure see measure

Grassman manifold see manifold

group 2,14 action 3 commutator C(n,H) 5,31 compact 34,117 connected 18 factorization 15 general affine GA(n) 4,56

positive GA+(n) 4

general linear GL(n) 3,15,18,19,34,38,56 positive GL+(n) 4

of isometries 51 isotropy (or isotropic) 4,13,35,36,38,44,46,80,83,108

instances 39,40,116 location - scale 5,31,40,51 opposite 14,37 orthogonalO(p) 5,7,17,38

special SO(p) 8,17,22,39 positive multiplicative 4 pseudo-orthogonal O(p,q) 7,16,19,26

special SO(p,q) 16,39,49,65

special,identity component sol (p,q) 7,17,23,26,39,63 quotient 37 representation 129 special linear SL(n) 115,118 special triangular ST(2) 118

140

standard transformation 12,13,28,43,48,49,83,117 sub- 10,52

closed 10,16,117,121,127 normal 37,47,91 regular 30,80

supplementary 49 symmetric 2,6 topological 2 triangular T+(n) 34,38,114

with unit diagonal T+ 1 (n) 20

unimodular 34 instances 39,40,41,56,120

see also factorization: Lie

Haar measure 32 see also invariant measure (left: right)

Hankel function 111

homogeneous space 3

hyperbolic distributions see conical surface model length 8

hyperboloid model 24,104,110,126 generalized 26,39,49,51,64 unit 8,23,39,126

hypergeometric function of matrix argument 105

independence 94,98,102 see also conditional

index parameter 106,108,110

invariance characterization by 43,49-50,52,65-66 subgroups 16,60-63

invariant measure 1,29,31,36,38,43,44,48,50,53,72,74,80,108,129 construction 53,55,66,72 and Jacobians 54-55,56,57,69 instances 31,38,39,40,49,51,56,57,58,59,62,63,65,66,67,73,77,84,85,

86,87,97,98,100,102,103,120,125,127,132 on cone 65,102 on cone surface 63 on cosets 52 on G(p,n) 59-60 on GL(p) 56

on gl(p,n)t 60

on Hq 39,85

on HP,q 51,63,65,72-74 on invariance subgroups 60-61

orbits of - - 61-63 on PD(n) 58-59 on Sp-1 62

141

under action of C(n,H) 31 under action of GA+(1) 31,40,84,85

left 32,37,52,55,59,120 on GA(n) 57 on GA+(1) 40,127

on T+(n) 57

left~right formulae 32 right 32,46,49,52,55,120 on GA(n) 57 on GA+(1) 40,127

on T+(n) 58

see also exterior product: differential: relatively

Iwasawa decomposition 27 of so(1,q) 25

Jacobi identity 17

Jacobian 45,54,69-70

Jordan normal form 118

Kronecker delta 123

LCD - space 12,13,50,83

Lie algebra 15,17,20,64

of GL(k), i.e. gl(k) 18,19 of O(p,q), i.e. o(p,q) 19-20 of SL(n), i.e. sl(n) 118 of SO(p), Le. so(p) 22

of sol (l,q), Le. so(l,q) 23 of ST(2), Le. st(2) 118 of T+1 (k), i.e. t+1 (k) 21

subalgebra 17 of gl (k) 19

Lie group 15,37,66,72,127

factorization 21 semisimple 27 subgroup 15-16,19

of GL(k) 19

Lie product 17 instance 18

likelihood function 107,110,112 marginal (function) 75,107,110

instance 111

likelihood ratio statistic 94

locally compact 12

location - scale model 75,82,83-85,93-94 see also group

Lorentz group 24 transformation 24

manifold Grassman G(p,n) 9,39 Riemannian 45,53,122 stiefel st(p,n) 9,39,98,105 see also differentiable

marginal distribution

formula 107 see also likelihood

marginalization principle 110

maximal invariant function 4,79,113

instances 51,67,104,105

142

statistic 14,75,90,107,110 instances 102,103,104,105,111,132

maximum likelihood estimator 82,94,128,131,132 existence and uniqueness 85

mean direction 76

metric Riemannian 45,122,123,125

modular function 32,34,52,108,120 in terms of Jacobians 55 of GA(n) 57 of T+(n) 34,58

of location-scale group 40

modulator 30,31,33,47,49,53,54 with multiplier 30,31,33,47-48,49,52

existence 30,30-31 instances 31,102

with quasi-multiplier 44,54-55 in terms of Jacobian 55

module see modular function

multiplier instances

normal

29,30,31,33,36,46,49 31,32,47,50,52,59,65,86,87,100,102,121

mUltivariate - distribution 6,76,78,79,121,125,131 - model 6-7,98

see also group

normalizer 47

143

orbit 3 types 106,116,119 constant - type 7,13-14,119

instances 7,8,9,118 regular - type 30,46,48,53,55,80 of invariance subgroup 61-63 projection 3,43,45,102 representative 4,7,14,48,79,84,88,116 space 3,48

orbital decomposition 4,7,13,14,44,54,55 instances 5,64-65,88,97,99,114,115,119,130

parametric statistics 1,74

parametrization (local) 15,45,122,123,124 orthonormal 123,124,125

polar decomposition 66

proper action mapping

quasi-

see action 13,29,48,50,83

invariant measure 35,43,53,54,121 multiplier 34-35,43,44,53,54,121

quotient measure 46,49

characterization by invariance 49,65 instances 50,51,52

topology 3

Radon measure 12,28

regular measure 12,28

relatively invariant measure 29,30,31,46 construction 31 instances 31,32,47,50,52,59,65,86,100,102,121 existence 30,36 uniqueness 30,36

relativity theory 24

Riemannian geometry 53 manifold see manifold metric 45

skew-symmetry 71

statistical inference 110

stereology 10,60

stiefel manifold see manifold

student distribution multivariate 79

submanifold 45,123,126

sUfficiency principle 82

sufficient statistic 90,102 minimal 82,105,111,129

transformation of densities 29,34 standard - group see group

transformation model 74,78,112 instances 105,106,131

balanced 82,112 exponential 2,75,128 main theorem 89-91 regularity conditions 83,89,91 standard 75,80,89,129

instances 85,87,89,95 see also composite

translation left 10 right 11

von Mises-Fisher model 76-78,87-89,95 matrix model 105-106

wedge product see exterior

Wishart distribution 94,128

1M

Notation index

a,aG 32 gx 3

s4 74 gK 7

gP 74

B(q) 24 9JL 29

~'~G 32 G 2

~ 28 GO 14

G(p,n) 9

C(n,H) 5 GA(n) 4

Cp(f,K) 76 GA+(n) 4

l( 29, 35 GL(n) 3

GL+ (n) 4

det 8 Gx 3

D 45 G 4 x

D,D G 10 G/K 7

A,A G 32 G\~ 3

" 3

Eix 45, 122 r 12

6,6H 10 r (g) 54

'Il 21

fir 69

'"(11) 29 HP,q 26

Hq 8

gl(k) 17 H\G 11

gl(p,n) 8 ':It 21

gl(p,n)t 9

gl+(2) 115 i 125

gl (2) 115 I p,q 16

gp 81 I 1 ,q 7

146

J 45 a "'/r 69

til 45

:t (~) 28 ~ 74

L 107 q 45

LCD 12

L 10 Rg 11 g

~ 67 * IR+ 4

m,ml( 30, 44 sign 69

Jl (f) 28 sl (n) 118 Jll( 30 seep) 22

Jl//3 46 se(l,q) 23

.M 75 st(2) 118

.M(~) 28 S (n) 58 Sp-1 22

Nq(f ,};) 76 St(p,n) 9

v 12 SL(n) 115, 118 J{ 78 SO(p,q) 16 p

SOl(p,q) 17

e(p) 20 sol (l,q) 7

e(p,q) 20 SO(q) 8

o (n) 5 ST(2) 118

O(p,q) 16 Y' (n) 6

0(1, q) 7 Y'(~) 2

P(q) 25 t+1 (n) 21

PD(n) 38 tr 105

11" (x) 3 T+(n) 34

147

T+1 (k) 20 1\ 69

TA+(P) 95 77

TMp 68 77

* TM P

68 < > 81

u(x) 4 - 64, 82

-x 5, 84

X+ 78

!: 2

!: 41-42 11"

z(x) 4

C ,CG 56

* (transposition)

* (product) 8

< > 5

II II 5

I I 8

[ , ] (element af

GA(n) ) 4

[ , ] (Lie multi-

plication) 17

81 21

18 28

0 29

et (product of measures) 37

et (tensor product) 76

Lecture Notes in Statistics Vol. 44: D.L. McLeish, Christopher G. Small, The Theory and Applications of Statistical Inference Functions. 136 pages, 1987.

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