an:01756306
Zbl 0995.60022
Ivanova, N. L.
The reconstruction of natural exponential families by their marginals
EN
J. Math. Sci., New York 106, No. 1, 2672-2681 (2001).
00080176
2001
j
60E99 62F10
Morris natural exponential family; cumulant function; bivariate exponential family of distributions; hyperbolic cosine distribution
Two-dimensional natural exponential families of distributions with cumulant function \(k(\theta_1,\theta_2)\) are considered. It is shown that the following relations hold
\[
\begin{aligned} k(\theta_1,\theta_2) &= k_1(\theta_1+\beta_1(\theta_2))+k_2(\theta_2)- k_1(\theta_1^0+\beta_1(\theta_2))\\ &= k_2(\theta_2+\beta_2(\theta_1))+k_1(\theta_1)- k_1(\theta_2^0+\beta_2(\theta_1)), \end{aligned}
\]
where \(k_1\) and \(k_2\) are the cumulant functions of the marginal distributions, \(\beta_1\) and \(\beta_2\) are some functions. Using marginals from the Morris class (i.e. the families in which the variance \(V\) is a quadratic function of the mean \(m\): \(V=Am^2+Bm+C\)) the author describes possible functions \(\beta_1\) and \(\beta_2\) and corresponding two-dimensional exponential families.
R.E.Maiboroda (Ky??v)