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In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a ...
The lognormal and Beta distribution are in the exponential family, but not the natural exponential family. The gamma distribution with two parameters is an exponential family but not a NEF and the chi-squared distribution is a special case of the gamma distribution with fixed scale parameter, and thus is also an exponential family but not a NEF ...
Download as PDF; Printable version; In other projects ... also known as the Natural Exponential Family ... (PDF). Biometrika. 81 (2): 396–402.
This page was last edited on 7 December 2016, at 21:07 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.
Download as PDF; Printable version; ... for this family. If the pdf for (,) is (;,), then the pdf ... is an exponential family with natural parameters ...
These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF). More general distributions can be defined by fixing some initial values of p j and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data.
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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
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