Page 308 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 291

Solving the above equations simultaneously, the MLEs of m and 2 are found
to be

n
1 X
^
1 ˆ x j ;
n
jˆ1
and
n
1 X 2
^
^
2 ˆ x j 1 † :
n
jˆ1
The maximum likelihood estimators for m and 2 are, therefore,
n 9
1
^
X
1 ˆ X j ˆ X; >
>
n >
>
jˆ1 >
=
9:106†
n
>
^
2
>
2 ˆ 1 X X j X† ˆ n 1 S ; >
2 >
>
n n ;
jˆ1
^
which coincide with their moment estimators in this case. Although 2 is
^
^
biased, consistency and asymptotic efficiency for both 1 and 2 can be easily
verified.
Example 9.16. Let us determine the MLE of considered in Example 9.12.
Now,
1
8
< ; for 0 x ;
f x; †ˆ 9:107†
0; elsewhere.
:
The likelihood function becomes
n
1
L x 1 ; x 2 ; ... ; x n ; †ˆ ; 0 x i ; for all i: 9:108†

A plot of L is given in Figure 9.5. However, we note from the condition
associated with Equation (9.108) that all sample values x i must be smaller than

or equal to , implying that only the portion of the curve to the right of
max (x 1 , . .., x n ) is applicable. Hence, the maximum of L occurs at
ˆ max (x 1 , x 2 ,..., x n ), or, the MLE for is
^
ˆ max x 1 ; x 2 ; ... ; x n †; 9:109†

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