Questions tagged [exponential-family]

In probability and statistics, the exponential family is an important class of probability distributions sharing a certain form, specified below. This special form is chosen for mathematical convenience, on account of some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural distributions to consider. The term exponential class is sometimes used in place of "exponential family".

From Mood et al. (pages 312 and 313, 1974):

1. One-parameter exponential family.

A one-parameter family ($\theta$ is unidimensional) of densities $f(.;\theta)$ that can be expressed as:

$f_X(x;\theta) = \text{a}(\theta)\text{b}(x)\text{exp}[\text{c}(\theta)\text{d}(x)]$,

for $-\infty < x < \infty$, for all $\theta \in$ parametric space, and for a suitable choice of functions $\text{a}(.),\text{b}(.),\text{c}(.)$, and $\text{d}(.)$ is defined to belong to the exponential family or exponential class.

2. K-parameter exponential family.

A family of densities $f(.,\theta_1,...,\theta_k)$ that can be expressed as:

$f_X(x;\theta_1,...,\theta_k) = \text{a}(\theta_1,...,\theta_k)\text{b}(x)\text{exp}\sum\limits_{j=1}^k{\text{c}_j(\theta_1,...,\theta_k)\text{d}_j(x)}$,

for a suitable choice of functions $\text{a}(.,...,.), \text{b}(.), \text{c}_j(.,...,.)$, $\text{d}_j(.)$, $j=1,....,k$, is defined to belong to the exponential family.

References:

Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to theory of statistics. (B. C. Harrinson & M. Eichberg, Eds.) (3rd ed., p. 564). McGraw-Hill, Inc.

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