ln L(q | X) = a(X) + b(q) + c1(X)s1(q) + c2(X)s2(q) + . . . + cK(X)sK(q)
where a(), b(), and cj() and sj() 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 cj()s and sj()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 cj() 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:
