In probability and statistics, the class of exponential dispersion models (EDM), also called exponential dispersion family (EDF), is a set of probability distributions that represents a generalisation of the natural exponential family. Exponential dispersion models play an important role in statistical theory, in particular in generalized linear models because they have a special structure which enables deductions to be made about appropriate statistical inference.

Definition

Univariate case

There are two versions to formulate an exponential dispersion model.

Additive exponential dispersion model

belongs to the additive exponential dispersion model, if its probability density function can be written as

Reproductive exponential dispersion model

is called reproductive exponential dispersion model, and is given by

. The terminology dispersion model as dispersion parameter is a natural exponential family.

Multivariate case

In the multivariate case, the n has a probability density function of the following form

.

Properties

Cumulant-generating function

The cumulant-generating function is given by

Mean and variance

are given by

.

Reproductive

are i.i.d. with

are called reproductive.

Unit deviance

The probability density function can also be expressed in terms of the unit deviance as

.

Examples

Many very common probability distributions belong to the class of EDMs, among them are: normal distribution, binomial distribution, Poisson distribution, negative binomial distribution, gamma distribution, inverse Gaussian distribution, and Tweedie distribution.

References