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On polynomial variance functions
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  • Published: March 1992

On polynomial variance functions

  • Shaul K. Bar-Lev1,
  • Daoud Bshouty2 &
  • Peter Enis3 

Probability Theory and Related Fields volume 94, pages 69–82 (1992)Cite this article

  • 138 Accesses

  • 14 Citations

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Summary

Let ℱ be a natural exponential family onℝ and (V, Ω) be its variance function. Here, Ω is the mean domain of ℱ andV, defined on Ω, is the variance of ℱ. A problem of increasing interest in the literature is the following: Given an open interval Ω⊂ℝ and a functionV defined on Ω, is the pair (V, Ω) a variance function of some natural exponential family? Here, we consider the case whereV is a polynomial. We develop a complex-analytic approach to this problem and provide necessary conditions for (V, Ω) to be such a variance function. These conditions are also sufficient for the class of third degree polynomials and certain subclasses of polynomials of higher degree.

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Author information

Authors and Affiliations

  1. Department of Statistics, University of Haifa, 31905, Haifa, Israel

    Shaul K. Bar-Lev

  2. Faculty of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Daoud Bshouty

  3. Department of Statistics, State University of New York at Buffalo, 14214, Buffalo, NY, USA

    Peter Enis

Authors
  1. Shaul K. Bar-Lev
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  2. Daoud Bshouty
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  3. Peter Enis
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Cite this article

Bar-Lev, S.K., Bshouty, D. & Enis, P. On polynomial variance functions. Probab. Th. Rel. Fields 94, 69–82 (1992). https://doi.org/10.1007/BF01222510

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  • Received: 23 July 1991

  • Revised: 14 April 1992

  • Issue Date: March 1992

  • DOI: https://doi.org/10.1007/BF01222510

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Mathematics Subject Classification (1980)

  • 62E10
  • 60J30
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