Page 310 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 293
^
The mean and variance of are

Z n
^
Ef gˆ xf ^ x†dx ˆ ; 9:114†
0 n ‡ 1

Z n 2 " n #
^
2
varf gˆ x f ^ x†dx ˆ 2 : 9:115†
0 n ‡ 1 n ‡ 1† n ‡ 2†
^
We see that is biased but consistent.
2
Example 9.17. Let us now determine the MLE of ˆ r in Example 9.13. To

carry out this estimation procedure, it is now necessary to determine the pdf of
X given by Equation (9.77). Applying techniques developed in Chapter 5, we
can show that X is characterized by the Rice distribution with pdf given by (see
Benedict and Soong, 1967)
8 1=2 2
x x x ‡
>
< I 0 exp ; for x 0;
2
f X x; ; †ˆ 2 2 2 2 9:116†
>
0; elsewhere;
:
where I 0 is the modified zeroth-order Bessel function of the first kind.
Given a sample of size n from population X, the likelihood function takes the
form
n
Y 2
L ˆ f X x j ; ; †: 9:117†
jˆ1
2 ^
The MLEs of and , and , satisfy the likelihood equations
2 b
q ln L ˆ 0; and q ln L ˆ 0; 9:118†
q ^ q b 2

which, upon simplifying, can be written as
n
1 X x j I 1 y j †
1 ˆ 0; 9:119†
n ^ 1=2 I 0 y j †
jˆ1
and

!
n
1 1 X 2
^
b 2
ˆ x ; 9:120†
j
2 n
jˆ1

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