Fundamentals of Probability and Statistics for Engineers

Answer: let us denote^2 by. Then,

and

Hence, according to Equation (9.36), the CRLB for the variance of any
unbiased estimator for is 2^2 /n.
For S^2 , it has been shown in Section 9.1.2 that it is an unbiased estimator for
and that its variance is [see Equation (9.10)]

since when X is normally distributed. The efficiency of S^2 , denoted by
e(S^2 ), is thus

)

We see that the sample variance is not an efficient estimator for in this
case. It is, however, asymptotically efficient in the sense that e(S^2 1 as

Parameter Estimation 271

 

f…X;†ˆ

1

… 2 †^1 =^2

exp

X^2

2 



;

lnf…X;†ˆ

X^2

2 



1

2

ln 2;

qlnf…X;†

q

ˆ

X^2

2 ^2



1

2 

;

q^2 lnf…X;†

q^2

ˆ

X^2

^3

‡

1

2 ^2

;

E

q^2 lnf…X;†

q^2



ˆ



^3

‡

1

2 ^2

ˆ

1

2 ^2

:

 



varfS^2 gˆ

1

n

 4 

n 3

n 1

^4



ˆ

1

n

3 ^4 

n 3

n 1

^4



ˆ

2 ^4

n 1

ˆ

2 ^2

n 1

;

 4 ˆ 3 ^4

e…S^2 †ˆ

CRLB

var…S^2 †

ˆ

n 1

n

:



!

n!1.