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The reconstruction of natural exponential families by their marginals. (English) Zbl 0995.60022
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.

MSC:
60E99 Distribution theory
62F10 Point estimation
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