Fundamentals of Probability and Statistics for Engineers

The mean and variance of are

We see that is biased but consistent.

Ex ample 9. 17. Let us now determine the MLE of r^2 in Example 9.13. To
carry out this estimation procedure, it is now necessary to determine the pdf of
X giv en by Equation (9.77). Applying techniques developed in Chapter 5, we
can show that X is characterized by the Rice distribution with pdf gi ven by (see
Benedict and Soong,1967)

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

The MLEs of and and satisfy the likelihood equations

which, upon simplifying, can be written as

and

Parameter Estimation 293

^

Ef^gˆ

Z

0

xf^…x†dxˆ

n

n‡ 1

; … 9 : 114 †

varf^gˆ

Z

0

x

n

n‡ 1



 2

f^…x†dxˆ

n

…n‡ 1 †^2 …n‡ 2 †

"

^2 : … 9 : 115 †

^

ˆ

fX…x;;^2 †ˆ

x

^2

I 0

^1 =^2 x

^2



exp 

x^2 ‡

2 ^2



; forx 0 ;

0 ; elsewhere;

8

><

>:

… 9 : 116 †

Lˆ

Yn

jˆ 1

fX…xj;;^2 †: … 9 : 117 †

 ^2 ,^ ^2 ,

qlnL

q^

ˆ 0 ; and

qlnL

qb^2

ˆ 0 ; … 9 : 118 †

1

n^^1 =^2

Xn

jˆ 1

xjI 1 …yj†

I 0 …yj†

 1 ˆ 0 ; … 9 : 119 †

b^2 ˆ^1

2

1

n

Xn

jˆ 1

x^2 j^

!

; … 9 : 120 †

b