The method of maximum likelihood (the term first used by Fisher, 1922a) is a general method of estimating parameters of a population by values that maximize the likelihood (L) of a sample. The likelihood L of a sample of n observations x1, x2, ..., xn, is the joint probability function p(x1, x2, ..., xn) when x1, x2, ..., xn are discrete random variables. If x1, x2, ..., xn are continuous random variables, then the likelihood L of a sample of n observations, x1, x2, ..., xn, is the joint density function f(x1, x2, ..., xn).

Let L be the likelihood
of a sample, where L is a function of the parameters q1, q2, ...

Let q be an element of W. If W is an open interval, and if L(q) is differentiable and assumes a maximum on W, then the MLE will be a solution of the following equation: (dL(q))/dq = 0. For more information, see Mendenhall and Sincich (1984), Bain and Engelhardt (1989), and Neter, Wasserman, and Kutner (1989).

See also, Nonlinear Estimation and Variance Components and Mixed Model ANOVA/ANCOVA.