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234 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

2
2
EXAMPLE 7-9 Let X be normally distributed with mean and variance , where both and are
unknown. The likelihood function for a random sample of size n is
n 1 1 n
2
2
2
2
1
L1 , 2 q e 1x i 2 12 2 2 n 2 e 11 2 2 a x i 2 2
i 1
i 1 12 12 2
and
n 1 n
2
2
ln L1 , 2 ln12 2 2 a 1x 2 2

i
2 2 i 1
Now
2
ln L1 , 2 1 n
2 a 1x 2 0

i 1 i
2
ln L1 , 2 n 1 n
2
4 a 1x 2 0

i
2
1 2 2 2 2 i 1
The solutions to the above equation yield the maximum likelihood estimators
1 n
2
n a
ˆ X ˆ 1X X 2 2
i
i 1
Once again, the maximum likelihood estimators are equal to the moment estimators.
Properties of the Maximum Likelihood Estimator
The method of maximum likelihood is often the estimation method that mathematical statisti-
cians prefer, because it is usually easy to use and produces estimators with good statistical
properties. We summarize these properties as follows.

Properties of
the Maximum Under very general and not restrictive conditions, when the sample size n is large and
ˆ
Likelihood if is the maximum likelihood estimator of the parameter ,
Estimator
ˆ
ˆ
(1) is an approximately unbiased estimator for 3E1 2 4 ,
ˆ
(2) the variance of is nearly as small as the variance that could be obtained
with any other estimator, and
ˆ
(3) has an approximate normal distribution.

Properties 1 and 2 essentially state that the maximum likelihood estimator is approxi-
mately an MVUE. This is a very desirable result and, coupled with the fact that it is fairly easy
to obtain in many situations and has an asymptotic normal distribution (the “asymptotic”
means “when n is large”), explains why the maximum likelihood estimation technique is
widely used. To use maximum likelihood estimation, remember that the distribution of the
population must be either known or assumed.