By Ole E Barndorff-Nielsen, Preben Blaesild, Poul S. Eriksen

The current set of notes grew out of our curiosity within the learn of statistical transformation versions, particularly exponential transfor- mation versions. The latter category includes as exact instances all totally tractable versions for mUltivariate common observations. the idea of decomposition and invariance of measures offers crucial instruments for the research of transformation versions. whereas the main facets of that thought are handled in a few mathematical monographs, as a rule as a part of a lot broader contexts, we now have chanced on no unmarried account within the literature that is sufficiently finished for statistical pur- poses. This quantity goals to fill the distance and to point the usefulness of degree decomposition and invariance thought for the technique of statistical transformation types. through the paintings with those notes we've got benefitted a lot from discussions with steen Arne Andersson, J0rgen Hoffmann-J0rgensen and J0rgen Granfeldt Petersen. we're additionally very indebted to Jette Ham- borg and Oddbj0rg Wethelund for his or her eminent secretarial tips.

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**Example text**

3. GLCn)-invariant measure on PDCn). 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) if 10 = T+(n)O(n) € is a left factorization. This means that we just have to calculate the PD(n), determinant of the linear isomorphism f(T O): Sen) .... Sen) * 1 .... T01To. The invariant measure is then given by If(T) 1- 1 d1 where 1 = TT*.

Suppose in the following that (G,~) is a standard transformation group. 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) ii) iii) p~(x) € ~ ~ p~(f) J K-almost every ~(Gx) € ~(K) J f(x)~(x) G\~ ~ whenever f € ~ (~) for every J Gx . 1. The measure ~ is quasi-invariant with quasi-multiplier ~(g,x) if and only if for K-almost every ~(x) € G\~ the measure p~(x) ~(g,y), (g,y) is quasi-invariant on € Gx with quasi-multiplier G x Gx.

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) o is invariant. The function m is called a modulator with quasi-multiplier x. 3), for the construction of a modulator on the basis of an orbital decomposition and ultimately for the construction 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 Recall that if ~.