Natural exponential family

In probability and statistics, a natural exponential family (NEF) is a class of probability distributions that is a special case of an exponential family (EF). The natural exponential families (NEF) are a subset of the exponential families. A NEF is an exponential family in which the natural parameter η and the natural statistic T(x) are both the identity. A distribution in an exponential family with parameter θ can be written with probability density function (PDF) where and are known functions. A distribution in a natural exponential family with parameter θ can thus be written with PDF [Note that slightly different notation is used by the originator of the NEF, Carl Morris. Morris uses ω instead of η and ψ instead of A.] Suppose that , then a natural exponential family of order p has density or mass function of the form: where in this case the parameter A member of a natural exponential family has moment generating function (MGF) of the form The cumulant generating function is by definition the logarithm of the MGF, so it is 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, Bernoulli, and geometric distributions are special cases of the above five distributions. For example, 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 ), and the geometric distribution is a special case of the negative binomial distribution. Some exponential family distributions are not NEF.