Natural Exponential Family - Examples

Examples

The five most important univariate cases are:

  • normal distribution with known variance
  • Poisson distribution
  • gamma distribution with known shape parameter α (or k depending on notation set used)
  • binomial distribution with known number of trials, n
  • negative binomial distribution with known

These five examples – Poisson, binomial, negative binomial, normal, and gamma – are a special subset of NEF, called NEF with quadratic variance function (NEF-QVF) because the variance can be written as a quadratic function of the mean. NEF-QVF are discussed below.

Distributions such as the exponential, chi-squared, Rayleigh, Weibull, Bernoulli, and geometric distributions are special cases of the above five distributions. Many common distributions are either NEF or can be related to the NEF. For example: the chi-squared distribution is a special case of the gamma distribution. The Bernoulli distribution is a binomial distribution with n = 1 trial. The exponential distribution is a gamma distribution with shape parameter α = 1 (or k = 1 ). The Rayleigh and Weibull distributions can each be written in terms of an exponential distribution.

Some exponential family distributions are not NEF. The lognormal and Beta distribution are in the exponential family, but not the natural exponential family.

The parameterization of most of the above distributions has been written differently than the parameterization commonly used in textbooks and the above linked pages. For example, the above parameterization differs from the parameterization in the linked article in the Poisson case. The two parameterizations are related by, where λ is the mean parameter, and so that the density may be written as

for, so

, and

This alternative parameterization can greatly simplify calculations in mathematical statistics. For example, in Bayesian inference, a posterior probability distribution is calculated as the product of two distributions. Normally this calculation requires writing out the probability distribution functions (PDF) and integrating; with the above parameterization, however, that calculation can be avoided. Instead, relationships between distributions can be abstracted due to the properties of the NEF described below.

An example of the multivariate case is the multinomial distribution with known number of trials.

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