## The Annals of Statistics

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

### Natural Exponential Families with Quadratic Variance Functions

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345690

**Digital Object Identifier**

doi:10.1214/aos/1176345690

**Mathematical Reviews number (MathSciNet)**

MR642719

**Zentralblatt MATH identifier**

0498.62015

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

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

#### Citation

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