Abstract
Consider an exponential family F on the set of non-negative integers indexed by the parameter a. The cumulative distribution function of an element of F estimated on k is both a function of a and k. Assume that the derivative of this function with respect to a is the product of three things: a function of k, a function of a and the function a to the power k. We show that this assumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. Ultimately, the proofs rely on the fact that only Möbius functions preserve the cross ratio.
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References
Abramowitz M, Stegun I (1965) Handbook of mathematical functions. Dover, New York
Diaconis P, Ylvisaker D (1979) Conjugate priors for exponential families. Ann Stat 7:269–281
Johnson NL, Kemp AW, Kotz S (2005) Univariate discrete distributions. Wiley, New York
Katz I (1965) Unified treatment of a broad class of discrete probability distributions. Classical and contagious discrete distributions. Pergamon Press, Oxford, pp 175–182
Kendall M, Stuart A (1977) The advanced theory of statistics, vol 1, 4th edn. Macmillan Publishing Corporation, New York
Morris CN (1982) Natural exponential families with quadratic variance functions. Ann Stat 10:65–80
Wackerly D, Mendenhall W III, Scheaffer RL (2002) Mathematical statistics with applications, 6th edn. Duxbury, Pacific Grove
Wadsworth GP (1960) Introduction to probability and random variables. Mc Graw Hill, New York
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Letac, G. Cumulative distribution functions for the five simplest natural exponential families. Metrika 82, 891–902 (2019). https://doi.org/10.1007/s00184-019-00710-z
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DOI: https://doi.org/10.1007/s00184-019-00710-z