Solving the above equations simultaneously, the MLEs of m and^2 are found
to be
and
The maximum likelihood estimators for m and^2 are, therefore,
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,
The likelihood function becomes
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 xi must be smaller than
or equal to , implying that only the portion of the curve to the right of
max(x 1 ,...,xn) is applicable. Hence, the maximum of L occurs at
max(x 1 ,x 2 ,...,xn), or, the MLE for is
Parameter Estimation 291
^ 1 ^1
n
Xn
j 1
xj;
^ 2 ^1
n
Xn
j 1
xj^ 1 ^2 :
^ 1 ^1
n
Xn
j 1
XjX;
^ 2 ^1
n
Xn
j 1
XjX^2
n 1
n
S^2 ;
9
>>
>>
>=
>>
>>>
;
9 : 106
^
^ ^
f
x;
1
; for 0 x;
0 ; elsewhere.
8
<
:
9 : 107
L
x 1 ;x 2 ;...;xn;
1
n
; 0 xi;foralli:
9 : 108
^max
x 1 ;x 2 ;...;xn;
9 : 109