Page 289 - Fundamentals of Probability and Statistics for Engineers

272 Fundamentals of Probability and Statistics for Engineers

Example 9.4. Problem: determine the CRLB for the variance of any unbiased
estimator for in the lognormal distribution

8
1 1
2
> exp ln x ; for x 0; and > 0;
< 1=2
f x; †ˆ x 2 † 2
>
0; elsewhere:
:
Answer: we have
2
q ln f X; † 1 ln X
ˆ ‡ ;
q 2 2 2
2
2
q ln f X; † 1 ln X
ˆ ;
q 2 2 2 3
2
q ln f X; † 1 1
E ˆ ˆ :
q 2 2 2 3 2 2
2
It thus follows from Equation (9.36) that the CRLB is 2 /n.

Before going to the next criterion, it is worth mentioning again that, although
unbiasedness as well as small variance is desirable it does not mean that we should
discard all biased estimators as inferior. Consider two estimators for a parameter ,
^
^
^
1 and 2 , the pdfs of which are depicted in Figure 9.2(a). Although 2 is biased,
^
because of its smaller variance, the probability of an observed value of 2 being
closer to the true value can well be higher than that associated with an observed
^
^
value of 1 . Hence, one can argue convincingly that 2 is the better estimator of
the two. A more dramatic situation is shown in Figure 9.2(b). Clearly, based on a
^
particular sample of size n, an observed value of 2 will likely be closer to the true
^
^

value than that of 1 even though 1 is again unbiased. It is worthwhile for us to
reiterate our remark advanced in Section 9.2.1 – that the quality of an estimator
does not rest on any single criterion but on a combination of criteria.
Example 9.5. To illustrate the point that unbiasedness can be outweighed by
other considerations, consider the problem of estimating parameter in the
binomial distribution
k
p X k†ˆ 1 † 1 k ; k ˆ 0; 1: 9:43†
^
^
Let us propose two estimators, 1 and 2 , for given by
9
^
1 ˆ X; >
= 9:44†
^
2 ˆ nX ‡ 1 ; ;
>
n ‡ 2

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