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

Answer: we have

It thus fo llows from Equation (9.36) that the CRLB is 2^2 /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 F igure 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. H ence, 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 va lue 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 thatunbiasedness can be outweighed by
other considerations, consider the problem of estimating parameter in the
binomial distribution

Let us propose two estimators, 1 and 2 , for given by

272 Fundamentals of Probability and Statistics for Engineers



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