The Annals of Statistics

Natural Exponential Families with Quadratic Variance Functions: Statistical Theory

Carl N. Morris

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The normal, Poisson, gamma, binomial, negative binomial, and NEFGHS distributions are the six univariate natural exponential families (NEF) with quadratic variance functions (QVF). This sequel to Morris (1982) treats certain statistical topics that can be handled within this unified NEF-QVF formulation, including unbiased estimation, Bhattacharyya and Cramer-Rao lower bounds, conditional distributions and moments, quadratic regression, conjugate prior distributions, moments of conjugate priors and posterior distributions, empirical Bayes and $G_2$ minimax, marginal distributions and their moments, parametric empirical Bayes, and characterizations.

Article information

Ann. Statist. Volume 11, Number 2 (1983), 515-529.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 60E05: Distributions: general theory
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F10: Large deviations 62E15: Exact distribution theory 62E30

Exponential families natural exponential families quadratic variance function normal distribution Poisson distribution gamma distribution binomial distribution negative binomial distribution NEG-GHS distribution unbiased estimation Bhattacharyya bounds quadratic regression conjugate priors Bayesian analysis posteriror moments $G_2$ minimax parametric empirical Bayes and characterizations


Morris, Carl N. Natural Exponential Families with Quadratic Variance Functions: Statistical Theory. Ann. Statist. 11 (1983), no. 2, 515--529. doi:10.1214/aos/1176346158.

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