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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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