Page 287 - Fundamentals of Probability and Statistics for Engineers

270 Fundamentals of Probability and Statistics for Engineers

which is a one-to-one transformation and differentiable with respect to ;
then,
2
dg †
^
^
CRLB for varf gˆ ‰CRLB for var †Š; 9:42†
d
^
where is an unbiased estimator for .
^
. Remark 4: given an unbiased estimator for parameter , the ratio of its

^
CRLB to its variance is called the efficiency of . The efficiency of any
unbiased estimator is thus always less than or equal to 1. An unbiased
estimator with efficiency equal to 1 is said to be efficient . We must point
out, however, that efficient estimators exist only under certain conditions.
We are finally in the position to answer the question posed in Example 9.2.
Example 9.2. part 2. Answer: first, we note that, in order to apply the CRLB,

pdf f (x; ) of population X must be known. Suppose that f (x; m) for this
example is N(m, 2 ). We have
" 2 #)
1 X m†
ln f X; m†ˆ ln 1=2 exp 2
2 † 2
" # 2
1 X m†
ˆ ln 1=2 2 ;
2 † 2
and

qlnf X; m† X m
ˆ :
qm 2
Thus,

)
qlnf X; m† 1 2 1
2
E ˆ Ef X m† gˆ :
qm 4 2
Equation (9.26) then shows that the CRLB for the variance of any unbiased
2

estimator for m is 2 /n. Since the variance of X is /n, it has the minimum
variance among all unbiased estimators for m when population X is distributed
normally.

Example 9.3. Problem: consider a population X having a normal distribution

N(0, 2 ) where 2 is an unknown parameter to be estimated from a sample of
size n > 1. (a) Determine the CRLB for the variance of any unbiased estimator
2
for 2 . (b) Is sample variance S an efficient estimator for 2 ?

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