Exponential dispersion model

Summary

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.[1][2][3] 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 edit

Univariate case edit

There are two versions to formulate an exponential dispersion model.

Additive exponential dispersion model edit

In the univariate case, a real-valued random variable   belongs to the additive exponential dispersion model with canonical parameter   and index parameter  ,  , if its probability density function can be written as

 

Reproductive exponential dispersion model edit

The distribution of the transformed random variable   is called reproductive exponential dispersion model,  , and is given by

 

with   and  , implying  . The terminology dispersion model stems from interpreting   as dispersion parameter. For fixed parameter  , the   is a natural exponential family.

Multivariate case edit

In the multivariate case, the n-dimensional random variable   has a probability density function of the following form[1]

 

where the parameter   has the same dimension as  .

Properties edit

Cumulant-generating function edit

The cumulant-generating function of   is given by

 

with  

Mean and variance edit

Mean and variance of   are given by

 

with unit variance function  .

Reproductive edit

If   are i.i.d. with  , i.e. same mean   and different weights  , the weighted mean is again an   with

 

with  . Therefore   are called reproductive.

Unit deviance edit

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

 

where the unit deviance takes the special form   or in terms of the unit variance function as  .

Examples edit

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 edit

  1. ^ a b Jørgensen, B. (1987). Exponential dispersion models (with discussion). Journal of the Royal Statistical Society, Series B, 49 (2), 127–162.
  2. ^ Jørgensen, B. (1992). The theory of exponential dispersion models and analysis of deviance. Monografias de matemática, no. 51.
  3. ^ Marriott, P. (2005) "Local Mixtures and Exponential Dispersion Models" pdf