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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.

A characterization of uniform distribution

Chachulska J.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 2

207-220

Is the Lebesgue measure on [0,1]2 a unique product measure on [0,1] 2 which is transformed again into a product measure on [0,1]2 by the mapping ψ(x, y) = (x, (x+y) mod 1))? Here a somewhat stronger version of this problem in a probabilistic framework is answered. It is shown that for independent and identically distributed random variables X and Y constancy of the conditional expectations of X+Y−I(X+Y>1) and its square given X identifies uniform distribution either absolutely continuous or discrete. No assumptions are imposed on the supports of the distributions of X and Y.