, Volume 3, Issue 2, pp 123-172

Exponential families with variance functions in \(\sqrt {\Delta P} (\sqrt \Delta )\) : Seshadri’s class: Seshadri’s class

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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.