A characterization of the Wishart exponential families by an invariance property
Tóm tắt
E is the space of real symmetric (d, d) matrices, andS and $$\bar S$$ are the subsets ofE of positive definite and semipositive-definite matrices. Let there be ap in $$\Lambda = \left\{ {\frac{1}{2},1,\frac{3}{2}, \ldots \frac{{d - 1}}{2}} \right\} \cup \left] {\frac{{d - 1}}{2}, + \infty } \right[$$ The Wishart natural exponential family with parameterp is a set of probability distributions on $$\bar S$$ defined by $$F_p = \{ \exp [ - \tfrac{1}{2}Tr(\Gamma x)](det\Gamma )^p \mu _p (dx);\Gamma \in S\} $$ where μp is a suitable measure on $$\bar S$$ . LetGL(ℝd) be the subset ofE of invertible matrices. Fora inGL(ℝd), define the automorphismg a ofE byg a(x)=t axa, where t a is the transpose ofa. The aim of this paper is to show that a natural exponential familyF onE is invariant byg a for alla inGL(ℝd) if and only if there existsp in Λ such that eitherF=F p, orF is the image ofF p byx↦−x. (Theorem).
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