Maximum Likelihood
Method
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, ... qk. Then
the maximum likelihood estimators
of q1, q2, ... qk are the
values of q1, q2, ... qk that maximize
L.
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.