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March, 1982 Natural Exponential Families with Quadratic Variance Functions
Carl N. Morris
Ann. Statist. 10(1): 65-80 (March, 1982). DOI: 10.1214/aos/1176345690


The normal, Poisson, gamma, binomial, and negative binomial distributions are univariate natural exponential families with quadratic variance functions (the variance is at most a quadratic function of the mean). Only one other such family exists. Much theory is unified for these six natural exponential families by appeal to their quadratic variance property, including infinite divisibility, cumulants, orthogonal polynomials, large deviations, and limits in distribution.


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Carl N. Morris. "Natural Exponential Families with Quadratic Variance Functions." Ann. Statist. 10 (1) 65 - 80, March, 1982.


Published: March, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0498.62015
MathSciNet: MR642719
Digital Object Identifier: 10.1214/aos/1176345690

Primary: 60E05
Secondary: 60E07 , 60F10 , 62E10 , 62E15 , 62E30

Keywords: Binomial distribution , Cumulants , exponential distribution , exponential families , gamma distribution , geometric distribution , hyperbolic secant distribution , Infinite divisibility , large deviations , limits in distribution , moments , natural exponential families , negative binomial distribution , normal distribution , orthogonal polynomials , Poisson distribution , quadratic variance function , variance function

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 1 • March, 1982
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