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, Volume 7, Issue 2, pp 391–412 | Cite as

Natural exponential families associated to Pick functions

  • Dhafer Malouche
Article

Abstract

We define a quadratic action of the group of invertible (2,2) matrices of determinant 1 by Mœbius transformsh(x)=(ax+b)/(cx+d) on the natural exponential families (NEF) on ℝ which changes the mean functionk′ of the NEFF in the new mean functionh(k′) associated to the new NEF, denoted byh(F). The variance function ofh(F) is(cm+d) 2 V F (h(m)). Whenzk′(z) orza k′ (a logz) happens to be a Pick function,h(F) can be explicitely described. We prove that certain cubic NEF belong to this type. This fact leads us to a classification of the variance functionsP(m)/m, where the polynomialP has degree ≤3 without complex zeros.

Key Words

Pick function variance function exponential families quadratic action 

AMS subject classification

32E20 60E07 60E10 62E10 

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Copyright information

© Sociedad de Estadística e Investigación Operativa 1998

Authors and Affiliations

  • Dhafer Malouche
    • 1
  1. 1.Laboratoire de Statistique et ProbabilitésUMR C55830Toulouse CedexFrance

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