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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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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DOI: https://doi.org/10.1007/BF01222510
Mathematics Subject Classification (1980)
- 62E10
- 60J30