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). Every distribution possessing a moment-generating function is a member of a natural exponential family, and the use of such distributions simplifies the theory and computation of generalized linear models.

The natural exponential family (NEF) is a subset of the exponential family. NEF is an exponential family in which the natural parameter η and the natural statistic T(x) are both the identity. A distribution in the exponential family with parameter θ can be written with probability density function (PDF)

where ${\displaystyle h(x)}$ and ${\displaystyle A(\theta )}$ are known functions. A distribution in the natural exponential family with parameter θ can thus be written with PDF

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