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). Every distribution possessing a momentgenerating function is a member of a natural exponential family, and the use of such distributions simplifies the theory and computation of generalized linear models.
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Transcription
Contents
Definition
Probability distribution function (PDF) of the univariate case (scalar domain, scalar parameter)
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.^{[1]} Morris uses ω instead of η and ψ instead of A.]
Probability distribution function (PDF) of the general case (multivariate domain and/or parameter)
Suppose that , then a natural exponential family of order p has density or mass function of the form:
where in this case the parameter
Moment and cumulant generating function
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
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 (NEFQVF) because the variance can be written as a quadratic function of the mean. NEFQVF are discussed below.
Distributions such as the exponential, chisquared, 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 chisquared 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 from 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
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.
Properties
The properties of the natural exponential family can be used to simplify calculations involving these distributions.
Univariate case
1. The cumulants of an NEF can be calculated as derivatives of the NEF's cumulant generating function. The nth cumulant is the nth derivative of the cumulant generating function with respect to t evaluated at t = 0.
The cumulant generating function is
The first cumulant is
The mean is the first moment and always equal to the first cumulant, so
The variance is always the second cumulant, and it is always related to the first and second moments by
so that
Likewise, the nth cumulant is
2. Natural exponential families (NEF) are closed under convolution.^{[citation needed]}
Given independent identically distributed (iid) with distribution from an NEF, then is an NEF, although not necessarily the original NEF. This follows from the properties of the cumulant generating function.
3. The variance function for random variables with an NEF distribution can be written in terms of the mean.^{[citation needed]}
4. The first two moments of a NEF distribution uniquely specify the distribution within that family of distributions.^{[citation needed]}
Multivariate case
In the multivariate case, the mean vector and covariance matrix are^{[citation needed]}
where is the gradient and is the Hessian matrix.
Natural exponential families with quadratic variance functions (NEFQVF)
A special case of the natural exponential families are those with quadratic variance functions. Six NEFs have quadratic variance functions (QVF) in which the variance of the distribution can be written as a quadratic function of the mean. These are called NEFQVF. The properties of these distributions were first described by Carl Morris.^{[2]}
The six NEFQVFs
The six NEFQVF are written here in increasing complexity of the relationship between variance and mean.
1. The normal distribution with fixed variance is NEFQVF because the variance is constant. The variance can be written , so variance is a degree 0 function of the mean.
2. The Poisson distribution is NEFQVF because all Poisson distributions have variance equal to the mean , so variance is a linear function of the mean.
3. The Gamma distribution is NEFQVF because the mean of the Gamma distribution is and the variance of the Gamma distribution is , so the variance is a quadratic function of the mean.
4. The binomial distribution is NEFQVF because the mean is and the variance is which can be written in terms of the mean as
5. The negative binomial distribution is NEFQVF because the mean is and the variance is
6. The (not very famous) distribution generated by the generalized^{[clarification needed]} hyperbolic secant distribution (NEFGHS) has^{[citation needed]} and
Properties of NEFQVF
The properties of NEFQVF can simplify calculations that use these distributions.
1. Natural exponential families with quadratic variance functions (NEFQVF) are closed under convolutions of a linear transformation.^{[citation needed]} That is, a convolution of a linear transformation of an NEFQVF is also an NEFQVF, although not necessarily the original one.
Given independent identically distributed (iid) with distribution from a NEFQVF. A convolution of a linear transformation of an NEFQVF is also an NEFQVF.
Let be the convolution of a linear transformation of X. The mean of Y is . The variance of Y can be written in terms of the variance function of the original NEFQVF. If the original NEFQVF had variance function
then the new NEFQVF has variance function
where
2. Let and be independent NEF with the same parameter θ and let . Then the conditional distribution of given has quadratic variance in if and only if and are NEFQVF. Examples of such conditional distributions are the normal, binomial, beta, hypergeometric and geometric distributions, which are not all NEFQVF.^{[1]}
3. NEFQVF have conjugate prior distributions on μ in the Pearson system of distributions (also called the Pearson distribution although the Pearson system of distributions is actually a family of distributions rather than a single distribution.) Examples of conjugate prior distributions of NEFQVF distributions are the normal, gamma, reciprocal gamma, beta, F, and t distributions. Again, these conjugate priors are not all NEFQVF.^{[1]}
4. If has an NEFQVF distribution and μ has a conjugate prior distribution then the marginal distributions are wellknown distributions.^{[1]}
These properties together with the above notation can simplify calculations in mathematical statistics that would normally be done using complicated calculations and calculus.
References
 Morris C. (1982) Natural exponential families with quadratic variance functions: statistical theory. Dept of mathematics, Institute of Statistics, University of Texas, Austin.