Applied Statistics and Probability for Engineers

7-3 METHODS OF POINT ESTIMATION 235

EXAMPLE 7-10 In the normal distribution case, the maximum likelihood estimators of and ^2 were

and. To obtain the maximum likelihood estimator of the function

, substitute the estimators and into the function h, which yields

Thus, the maximum likelihood estimator of the standard deviation is notthe sample

standard deviation S.

Complications in Using Maximum Likelihood Estimation

While the method of maximum likelihood is an excellent technique, sometimes complications

arise in its use. For example, it is not always easy to maximize the likelihood function because

the equation(s) obtained from may be difficult to solve. Furthermore, it may

not always be possible to use calculus methods directly to determine the maximum of L().

These points are illustrated in the following two examples.

EXAMPLE 7-11 Let Xbe uniformly distributed on the interval 0 to a. Since the density function is

for 0xaand zero otherwise, the likelihood function of a random sample of size nis

L 1 a 2 q

n

i 1

1

a

1

an

f 1 x 2  (^1) a
dL 1  (^2) d 0
ˆ 2 ˆ^2 c
1
n^ a
n
i 1
1 XiX 22 d
(^1)  2
h 1 , ^22  2 ^2  ˆ ˆ^2
ˆ^2  g
n
i 1 1 XiX^22 n
ˆX
To illustrate the “large-sample” or asymptotic nature of the above properties, consider the
maximum likelihood estimator for ^2 , the variance of the normal distribution, in Example 7-9.
It is easy to show that
The bias is
Because the bias is negative, tends to underestimate the true variance ^2. Note that the
bias approaches zero as nincreases. Therefore, is an asymptotically unbiased estimator
for ^2.
We now give another very important and useful property of maximum likelihood
estimators.
ˆ^2
ˆ^2
E 1 ˆ^22 ^2 
n 1
n^ 
(^2)  (^2) 
2
n
E 1 ˆ^22 
n 1
n^ 
2
Let be the maximum likelihood estimators of the parameters  1 ,
 2 ,, k. Then the maximum likelihood estimator of any function h( 1 ,  2 ,, k)
of these parameters is the same function of the estimators
ˆ 1 , ˆ 2 ,p, ˆk.
h 1 ˆ 1 , ˆ 2 ,p, ˆk 2
pp
ˆ 1 , ˆ 2 ,p, ˆk
The Invariance
Property
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