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, Volume 3, Issue 2, pp 123–172 | Cite as

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

  • Célestin C. Kokonendji
Article

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.

Keywords

Branching processes Infinitely divisible measures Lagrange expansion and distributions Natural exponential families Power and modified power series distributions Random graphs Variance functions 

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Copyright information

© SEIO 1994

Authors and Affiliations

  • Célestin C. Kokonendji
    • 1
  1. 1.Laboratoire de Statistique et ProbabilitésUniversité Paul Sabatier-CNRS U.R.A. D0745Toulouse cedexFrance

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