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Characteristic property of a class of multivariate variance functions

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Abstract

Natural exponential families (NEFs) are well known to be characterized by their variance functions. A problem of increasing interest for dimension d > 1 is the following: given an open convex set Ω of (0,∞)d and a real analytic function V from Ω into the set of linear symmetric operators from ℝd, is V a variance function of some NEF? In the real line case of d = 1, this question was already solved. The aim of this work is to give necessary and sufficient conditions on V in order to be the variance function for some multivariate NEF. The notion of absolutely monotonic function on [0,∞)d is thus introduced, and the determination of moments of the NEF is also involved. For an NEF concentrated on [0,∞)d, a bridge is established between the behavior of V around of the origin and the existence conditions of the corresponding NEF. Some illustrating examples are presented.

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Authors and Affiliations

  1. Laboratoire de Physique-Mathématiques, Université de Sfax, Faculté des Sciences, B.P. 1171, Sfax, Tunisie

    Abdelaziz Ghribi

  2. Laboratoire de Mathématiques de Besançon - UMR 6623 CNRS-UFC, Université de Franche-Comté, 16 route de Gray, 25030, Besançon Cedex, France

    Célestin C. Kokonendji

  3. Laboratoire de Probabilités et Statistique, Université de Sfax, Faculté des Sciences, B.P. 1171, Sfax, Tunisie

    Afif Masmoudi

Authors
  1. Abdelaziz Ghribi
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  2. Célestin C. Kokonendji
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  3. Afif Masmoudi
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Correspondence to Abdelaziz Ghribi.

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Ghribi, A., Kokonendji, C.C. & Masmoudi, A. Characteristic property of a class of multivariate variance functions. Lith Math J 55, 506–517 (2015). https://doi.org/10.1007/s10986-015-9295-7

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  • DOI: https://doi.org/10.1007/s10986-015-9295-7

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