Let f(x; ) be the density function of population X where, for simplicity, is
the only parameter to be estimated from a set of sample values x 1 ,x 2 ,...,xn.
The joint density function of the corresponding sample X 1 ,X 2 ,...,Xn has the
form
We define the likelihood function L of a set of n sample values from the
population by
In the case when X is discrete, we write
When the sample values are given, likelihood function L becomes a function
of a single variable. The estimation procedure for based on the method of
maximum likelihood consists of choosing, as an estimate of , the particular
value of that maximizes L. The maximum of L( ) occurs in most cases at the
value of where dL( )/d is zero. Hence, in a large number of cases, the
maximum likelihood estimate (MLE)^ of based on sample values x 1 ,x 2 ,...,
and xn can be determined from
As we see from Equations (9.96) and (9.97), function L is in the form of a
product of many functions of. Since L is always nonnegative and attains its
maximum for the same value of^ as ln L, it is ge ne rally easier to obtain MLE
by solving
because ln L is in the form of a sum rather than a product.
Equation (9.99) is referred to as the likelihood equation. The desired solution
is one where root is a function of xj,j 1, 2,...,n,ifsucharootexists.When
several roots of Equation (9.99) exist, the MLE is the root corresponding to the
global maximum of L or ln L.
288 Fundamentals of Probability and Statistics for Engineers
f
x 1 ;f
x 2 ;f
xn;:L
x 1 ;x 2 ;...;xn;f
x 1 ;f
x 2 ;f
xn;:
9 : 96 L
x 1 ;x 2 ;...;xn;p
x 1 ;p
x 2 ;p
xn;:
9 : 97 ^
dL
x 1 ;x 2 ;...;xn;^
d^ 0 : 9 : 98
^
^
dlnL
x 1 ;...;xn;^
d^