Lecture Notes in Statistics ----------------------------------------------------------------------.- VoL 1: RA Fisher: An Appreciation_ Edited by S.E. Fien- berg and D.V. Hinkley. XI, 208 pages, 1980. VoL 2: Mathematical Statistics and Probability Theory. Pro- ceedings 1978. Edited by W. Klonecki, A. Kozek, and J. Rosiriski. XXIV, 373 pages, 1980. Vol. 3: B.D. Spencer, Benefit-Cost Analysis of Data Used to Allocate Funds. VIII, 296 pages, 1980. VoL 4: E.A. van Doorn, Stochastic Monotonicity and Queueing Applications of Birth-Death Processes. VI, 118 pages, 1981. Vol. 5: T Rolski, Stationary Random Processes Asso- ciated with Point Processes. VI, 139 pages, 1981. Vol. 6: S.S. Gupta and D.-y' Huang, Multiple Statistical Decision Theory: Recent Developments. VIII, 104 pages, 1981. VoL 7: M. Akahira and K. Takeuchi, Asymptotic Efficiency of Statistical Estimators. VIII, 242 pages, 1981. Vol. 8: The First Pannonian Symposium on Mathematical Statistics. Edited by P Revesz, L. Schmetterer, and V.M. Zolotarev. VI, 308 pages, 1981. Vol. 9: B. Jorgensen, Statistical Properties of the Gen- eralized Inverse Gaussian Distribution. VI, 188 pages, 1981. Vol. 10: A.A. Mcintosh, Fitting Linear Models: An Ap- plication on Conjugate Gradient Algorithms. VI, 200 pages, 1982. VoL 11: D.F Nicholls and B.G. Quinn, Random Coefficient Autoregressive Models: An Introduction. V, 154 pages, 1982. Vol. 12: M. Jacobsen, Statistical Analysis of Counting Pro- cesses. VII, 226 pages, 1982. Vol. 13: J. Pfanzagl (with the assistance of W. Wefel- meyer), Contributions to a General Asymptotic Statistical Theory. VII, 315 pages, 1982. Vol. 14: GUM 82: Proceedings of the International Con- ference on Generalised Linear Models. Edited by R. Gil- christ. V, 188 pages, 1982. Vol. 15: K.R.W. Brewer and M. Hanif, Sampling with Un- equal Probabilities. IX, 164 pages, 1983. VoL 16: Specifying Statistical Models: From Parametric to Non-Parametric, Using Bayesian or Non-Bayesian Approaches. Edited by J.P Florens, M. Mouchart, J.P Raoult, L. Simar, and A.FM. Smith, XI, 204 pages, 1983. VoL 17: IV Basawa and D.J. Scott, Asymptotic Optimal Inference for Non-Ergodic Models. IX, 170 pages, 1983. Vol. 18: W. Britton, Conjugate Duality and the Exponential Fourier Spectrum. V, 226 pages, 1983. Vol. 19: L. Fernholz, von Mises Calculus For Statistical Functionals. VIII, 124 pages, 1983. VoL 20: Mathematical Learning Models - Theory and Algorithms: Proceedings of a Conference. Edited by U. Herkenrath, Q. Kalin, W. VogeL XIV, 226 pages, 1983. VoL 21: H. Tong, Threshold Models in Non-linear Time Series Analysis. X, 323 pages, 1983. VoL 22: S. Johansen, Functional Relations, Random Coef- ficients and Nonlinear Regression with Application to Kinetic Data. VIII. 126 pag(?s, 1984. Vol. 23: D.G. Saphim. Estimation of Victimization Pre- valence Using Data from the National Crime Survey. V, 165 pages. 1984. Vol. 24: TS. Rao, M.M. Gabr, An Introduction to Bispectral Analysis and Bilinear Time Series Models. VIII, 280 pages, 1984. VoL 25: Time Series Analysis of Irregularly Observed Data. Proceedings, 1983. Edited by E. Parzen. VII, 363 pages, 1984. Vol. 26: Robust and Nonlinear Time Serios Analysis. Pro- ceedings, 1983. Edited by J. Franko, W. Hardie and D. Martin. IX. 286 pages, 1984. Vol. 27: A. Janssen, H. Milbrodt, H. Strasser. Infinitely Divisible Statistical Experiments. VI, 163 pa(Jes, 1985. Vol. 28: S. Amari, Diffemntia!-Geometrical Methods in Sta- tistics. V, 290 pa(Jes_ 1985. VoL 29: Statistics in Ornithqlogy. Edited by B.J.T and PM. North. XXV, 418 pages. 1985. Vol. 30: J. Grandell, Stochastic Models of Air Pollutant Concentration. V, 110 pages, 1985. VoL 31: J. Pfanzagl, Asymptotic Expansions for General Statistical Models. VII, 505 pa(Jes. 1985. Vol. 32: Guneralized Linear Modols. Proceedin(Js, 1985. Edited by R. Gilchrist, B. Francis and J. Whittaker. VL 178 pa(Jes. 1985. VoL 33: M. Csor(J6. S. 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Lecture Notes in Statistics ----------------------------------------------------------------------.-
VoL 1: RA Fisher: An Appreciation_ Edited by S.E. Fienberg and D.V. Hinkley. XI, 208 pages, 1980.
VoL 2: Mathematical Statistics and Probability Theory. Proceedings 1978. Edited by W. Klonecki, A. Kozek, and J. Rosiriski. XXIV, 373 pages, 1980.
Vol. 3: B.D. Spencer, Benefit-Cost Analysis of Data Used to Allocate Funds. VIII, 296 pages, 1980.
VoL 4: E.A. van Doorn, Stochastic Monotonicity and Queueing Applications of Birth-Death Processes. VI, 118 pages, 1981.
Vol. 5: T Rolski, Stationary Random Processes Associated with Point Processes. VI, 139 pages, 1981.
VoL 7: M. Akahira and K. Takeuchi, Asymptotic Efficiency of Statistical Estimators. VIII, 242 pages, 1981.
Vol. 8: The First Pannonian Symposium on Mathematical Statistics. Edited by P Revesz, L. Schmetterer, and V.M. Zolotarev. VI, 308 pages, 1981.
Vol. 9: B. Jorgensen, Statistical Properties of the Generalized Inverse Gaussian Distribution. VI, 188 pages, 1981.
Vol. 10: A.A. Mcintosh, Fitting Linear Models: An Application on Conjugate Gradient Algorithms. VI, 200 pages, 1982.
VoL 11: D.F Nicholls and B.G. Quinn, Random Coefficient Autoregressive Models: An Introduction. V, 154 pages, 1982.
Vol. 12: M. Jacobsen, Statistical Analysis of Counting Processes. VII, 226 pages, 1982.
Vol. 13: J. Pfanzagl (with the assistance of W. Wefelmeyer), Contributions to a General Asymptotic Statistical Theory. VII, 315 pages, 1982.
Vol. 14: GUM 82: Proceedings of the International Conference on Generalised Linear Models. Edited by R. Gilchrist. V, 188 pages, 1982.
Vol. 15: K.R.W. Brewer and M. Hanif, Sampling with Unequal Probabilities. IX, 164 pages, 1983.
VoL 16: Specifying Statistical Models: From Parametric to Non-Parametric, Using Bayesian or Non-Bayesian Approaches. Edited by J.P Florens, M. Mouchart, J.P Raoult, L. Simar, and A.FM. Smith, XI, 204 pages, 1983.
VoL 17: IV Basawa and D.J. Scott, Asymptotic Optimal Inference for Non-Ergodic Models. IX, 170 pages, 1983.
Vol. 18: W. Britton, Conjugate Duality and the Exponential Fourier Spectrum. V, 226 pages, 1983.
Vol. 19: L. Fernholz, von Mises Calculus For Statistical Functionals. VIII, 124 pages, 1983.
VoL 20: Mathematical Learning Models - Theory and Algorithms: Proceedings of a Conference. Edited by U. Herkenrath, Q. Kalin, W. VogeL XIV, 226 pages, 1983.
VoL 21: H. Tong, Threshold Models in Non-linear Time Series Analysis. X, 323 pages, 1983.
VoL 22: S. Johansen, Functional Relations, Random Coefficients and Nonlinear Regression with Application to Kinetic Data. VIII. 126 pag(?s, 1984.
Vol. 23: D.G. Saphim. Estimation of Victimization Prevalence Using Data from the National Crime Survey. V, 165 pages. 1984.
Vol. 24: TS. Rao, M.M. Gabr, An Introduction to Bispectral Analysis and Bilinear Time Series Models. VIII, 280 pages, 1984.
VoL 25: Time Series Analysis of Irregularly Observed Data. Proceedings, 1983. Edited by E. Parzen. VII, 363 pages, 1984.
Vol. 26: Robust and Nonlinear Time Serios Analysis. Proceedings, 1983. Edited by J. Franko, W. Hardie and D. Martin. IX. 286 pages, 1984.
Vol. 27: A. Janssen, H. Milbrodt, H. Strasser. Infinitely Divisible Statistical Experiments. VI, 163 pa(Jes, 1985.
Vol. 28: S. Amari, Diffemntia!-Geometrical Methods in Statistics. V, 290 pa(Jes_ 1985.
VoL 29: Statistics in Ornithqlogy. Edited by B.J.T Mor~Jan and PM. North. XXV, 418 pages. 1985.
Vol. 30: J. Grandell, Stochastic Models of Air Pollutant Concentration. V, 110 pages, 1985.
VoL 31: J. Pfanzagl, Asymptotic Expansions for General Statistical Models. VII, 505 pa(Jes. 1985.
Vol. 32: Guneralized Linear Modols. Proceedin(Js, 1985. Edited by R. Gilchrist, B. Francis and J. Whittaker. VL 178 pa(Jes. 1985.
VoL 33: M. Csor(J6. S. Csiir(Jo. L. Horv,ith, An Asymptotic Theory for Empirical Reliability and Concontration Processes. V. 1'71 pa(Jes, 1986.
Vol. 35: Linear Statistical Inference. Proceedings, 1984. Edited by T Caliriski and W. Kloll(-?cki. VI, 318 pa(Jes, 1985.
VoL 36: B. Matern. Spatial Variation. Second Edition. 151 pages, 1986.
Vol. 37: Advancr?s in Ordln Restricted Statistical Inference. Proceudin(Js, 1985. Edited by R. Dykstra, T Robertson and FT Wright. VIIL 295 pages. 1986.
Vol. 38: Survey Research Desi(Jns: Towards a Better Understanding of Their Costs and Benefits. Edited by R.W. Puarson and R.F Boruch. V, 129 pages, 1986.
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
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.
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
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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
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.
Vol. 45: J.K. Ghosh,.Statisticallnformation and Likelihood. 384 pages, 1988.
Vol. 51: J. Husler, R-D. Reiss (Eds.)' Extreme Value Theory. Proceedings, 1987. X, 279 pages, 1989.
Vol. 52: P.K. Goel, T. Ramalingam, The Matching Methodology: Some Statistical Properties. VIII, 152 pages, 1989.
Vol. 53: B.C. Arnold, N. Balakrishnan, Relations, Bounds and Approximations for Order Statistics. IX, 173 pages, 1989.
Vol. 54: K. R Shah, B. K. Sinha, Theory of Optimal Designs. VIII, 171 pages. 1989.
Vol. 55: L. McDonald, B. Manly, J. Lockwood, J. Logan (Eds.), Estimation and Analysis of Insect Populations. Proceedings, 1988. XIV, 492 pages, 1989.
Vol. 56: J.K. Lindsey, The Analysis of Categorical Data Using GLiM. V, 168 pages. 1989.
Vol. 57: A. Decarli, B.J. Francis, R Gilchrist, G.U.H. Seeber (Eds.), Statistical Modelling. Proceedings, 1989. IX, 343 pages. 1989.
Vol. 58: O. E. Barndorff-Nielsen, P. Blaasild, P. S. Eriksen, Decomposition and Invariance of Measures, and Statistical Transformation Models. V, 147 pages. 1989.