Page 288 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 271

Answer: let us denote 2 by . Then,
1 X 2
f X; †ˆ exp ;
2 † 1=2 2

and

X 2 1
ln f X; †ˆ ln 2 ;
2 2
q ln f X; † X 2 1
ˆ ;
q 2 2 2
2
q ln f X; † X 2 1
ˆ ‡ ;
q 2 3 2 2
2
q ln f X; † 1 1
E ˆ ‡ ˆ :
q 2 3 2 2 2 2
Hence, according to Equation (9.36), the CRLB for the variance of any
2
unbiased estimator for is 2 /n.

2
For S , 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)]

1 n 3 4
2
varfS gˆ 4
n n 1
1 n 3
4
ˆ 3 4
n n 1
2 4 2 2
ˆ ˆ ;
n 1 n 1

2
since 4 ˆ 3 4 when X is normally distributed. The efficiency of S , denoted by
2
e(S ), is thus
CRLB n 1
2
e S †ˆ ˆ :
2
var S † n
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!
n !1.

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