BIVARIATE NATURAL EXPONENTIAL FAMILIES WITH LINEAR
DIAGONAL VARIANCE FUNCTIONS

Abstract: 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.

2000 AMS Mathematics Subject Classification: Primary: 62E10, 44A10; Secondary:
60E05, 60E10.

Keywords and phrases: Natural exponential families, variance functions, Laplace
transforms.