Definition of Exponential Family

Definition: A distribution is a member of the exponential family of distributions if its log-likelihood function can be written in the form below.

ln L(q | X) = a(X) + b(q) + c₁(X)s₁(q) + c₂(X)s₂(q) + . . . + c_K(X)s_K(q)

where a(), b(), and c_j() and s_j() for each j=1 to K are functions; q is the vector of all parameters; X is the matrix of observable data; and L() is the likelihood function as defined by the maximum likelihood procedure.

The members of the exponential family vary from each other in a(), b(), and the c_j()s and s_j()s. Most common named distributions are members of the exponential family.

Quoting from Greene, 1997, page 149: "If the log-likelihood function is of this form, then the functions c_j() are called sufficient statistics [and] the method of moments estimators(s) will be functions of them," Those estimators will be the maximum likelihood estimators which are asymptotically efficient here. (Econterms)

Terms related to The Exponential Family:

• Maximum likelihood
• Sufficient statistics
• Method of moments

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