, Volume 3, Issue 2, pp 123–172

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

  • Célestin C. Kokonendji

DOI: 10.1007/BF02562698

Cite this article as:
Kokonendji, C.C. Test (1994) 3: 123. doi:10.1007/BF02562698


This paper presents a classification of the setS3 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.S3 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 ofS3. The final result gives the closure ofS3 under the topology of weak convergence.


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

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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