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Wishart Laws and Variance Function on Homogeneous Cones

Graczyk Piotr, Ishi Hideyuki, Kołodziejek Bartosz

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 2

337--360

We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.

Bivariate natural exponential families with linear diagonal variance functions

Chachulska J.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 1

121--139

It is well known that natural exponential families (NEFs) are uniquely determined by their variance functions (VFs). However, there exist examples showing that even an incomplete knowledge of a matrix VF can be sufficient to determine a multivariate NEF. Following such an idea, in this paper a complete description of bivariate NEFs with linear diagonal of the matrix VF is given. As a result we obtain the families of distributions with marginals that are some combinations of Poisson and normal distributions. Furthermore, the characterization extends (in two-dimensional case) the classification of NEFs with linear matrix VF obtained by Letac [11]. The main result is formulated in terms of regression properties.