Fundamentals of Probability and Statistics for Engineers

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 ˆ^1

n

Xn

jˆ 1

xj;

^ 2 ˆ^1

n

Xn

jˆ 1

…xj^ 1 †^2 :



^ 1 ˆ^1

n

Xn

jˆ 1

XjˆX;

^ 2 ˆ^1

n

Xn

jˆ 1

…XjX†^2 ˆ

n 1

n

S^2 ;

9

>>

>>

>=

>>

>>>

;

… 9 : 106 †

^

^ ^



f…x;†ˆ

1



; for 0 x;

0 ; elsewhere.

8

<

:

… 9 : 107 †

L…x 1 ;x 2 ;...;xn;†ˆ

1



n

; 0 xi;foralli: … 9 : 108 †



ˆ 

^ˆmax…x 1 ;x 2 ;...;xn†; … 9 : 109 †