Fundamentals of Probability and Statistics for Engineers

272 Fundamentals of Probability and Statistics for EngineersExample 9.4. Problem: determine the CRLB for the variance of any unbiasedestimator for in the lognormal distribution8 >0;>:0; elsewhere:Answer: we haveq ln f …X; †qq 2 ln f …X; †q 2 ˆ 12 2E q2 ln f …X; †q 2ˆ 12 ‡ ln2 X2 2 ;ˆ 12 2ln 2 X 3 ; 3 ˆ12 2 :It thus follows from Equation (9.36) that the CRLB is 2 2 /n.Before going to the next criterion, it is worth mentioning again that, althoughunbiasedness as well as small variance is desirable it does not mean that we shoulddiscard 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 beingcloser to the true value can well be higher than that associated with an observedvalue of ^ 1. Hence, one can argue convincingly that ^ 2 is the better estimator ofthe two. A more dramatic situation is shown in Figure 9.2(b). Clearly, based on aparticular sample of size n, an observed value of ^ 2 will likely be closer to the truevalue than that of ^ 1 even though ^ 1 is again unbiased. It is worthwhile for us toreiterate our remark advanced in Section 9.2.1 – that the quality of an estimatordoes not rest on any single criterion but on a combination of criteria.Example 9.5. To illustrate the point that unbiasedness can be outweighed byother considerations, consider the problem of estimating parameter in thebinomial distributionp X …k† ˆ k …1 † 1 k ; k ˆ 0; 1: …9:43†Let us propose two estimators, ^ 1 and ^ 2, for given by^ 1 ˆ X;^ 2 ˆ nX ‡ 1n ‡ 2 ; 9>=> ;…9:44†TLFeBOOK