The Annals of Statistics

Natural Exponential Families with Quadratic Variance Functions

Carl N. Morris

Full-text: Open access


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.

Article information

Ann. Statist. Volume 10, Number 1 (1982), 65-80.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60E05: Distributions: general theory
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F10: Large deviations 62E10: Characterization and structure theory 62E15: Exact distribution theory 62E30

Exponential families natural exponential families quadratic variance function normal distribution Poisson distribution gamma distribution exponential distribution binomial distribution negative binomial distribution geometric distribution hyperbolic secant distribution orthogonal polynomials moments cumulants large deviations infinite divisibility limits in distribution variance function


Morris, Carl N. Natural Exponential Families with Quadratic Variance Functions. Ann. Statist. 10 (1982), no. 1, 65--80. doi:10.1214/aos/1176345690.

Export citation