Summary
This paper presents a classification of the setS 3 of all natural exponential families (NEF) on ℝ which have a variance function of the form\(\sqrt {\Delta P} (\sqrt \Delta )\), whereP is a polynomial of degree 3 and Δ is an affine function of the mean of the NEF. Particular cases have been considered previosly by V. Seshadri and can be obtained by a Lindsay transform of the NEF with cubic variance, as classified by Marianne Mora.S 3 may be split into six types and we provide a probabilistic interpretation of each of them; in particular, we show that the literature on random mappings provides several examples of discrete elements ofS 3. The final result gives the closure ofS 3 under the topology of weak convergence.
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Kokonendji, C.C. Exponential families with variance functions in\(\sqrt {\Delta P} (\sqrt \Delta )\): Seshadri’s class: Seshadri’s class. Test 3, 123–172 (1994). https://doi.org/10.1007/BF02562698
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DOI: https://doi.org/10.1007/BF02562698