Applied Statistics and Probability for Engineers

234 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

EXAMPLE 7-9 Let Xbe normally distributed with mean and variance ^2 , where both and ^2 are

unknown. The likelihood function for a random sample of size nis

and

Now

The solutions to the above equation yield the maximum likelihood estimators

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.

ˆX ˆ^2 

1

n^ a

n

i 1

1 XiX 22

 ln L 1 , ^22

 1 ^22



n

2 ^2



1

2 ^4

(^) a
n
i 1
1 xi 22  0
 ln L 1 , ^22


1
^2
(^) a
n
i 1
1 xi 2  0
ln L 1 , ^22 
n
2
ln 12 ^22 
1
2 ^2
(^) a
n
i 1
1 xi 22
L 1 , ^22 q
n
i 1
1
 12 
e^1 xi^2
(^2)  12  (^22)

1
12 ^22 n^2
e^11 ^2 
(^22) an
i 1 1 xi^2
2
Under very general and not restrictive conditions, when the sample size nis large and
if is the maximum likelihood estimator of the parameter ,
(1) is an approximately unbiased estimator for ,
(2) the variance of is nearly as small as the variance that could be obtained
with any other estimator, and
(3) ˆ has an approximate normal distribution.
ˆ
ˆ  3 E 1 ˆ 2  4
ˆ
Properties of
the Maximum
Likelihood
Estimator
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 nis 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.
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