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

摘要

This paper presents a classification of the setS3of all natural exponential families (NEF) on ℝ which have a variance function of the form$$sqrt {Delta P} (sqrt Delta )$$, wherePis 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.S3may 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 ofS3. The final result gives the closure ofS3under the topology of weak convergen