Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 283Ex ample 9. 12. Suppose that population X has a uniform distribution over therange (0, ) and we wish to estimate parameter from a sample of size n.The density function of X is8< 1; for 0 x ;f …x; † ˆ …9:73†:0; elsewhere;and the first moment is 1 ˆ 2 : …9:74†It follows from the method of moments that, on letting 1 ˆ X, we obtain^ ˆ 2X ˆ 2 X nX j :njˆ1…9:75†Upon little reflection, the validity of this estimator is somewhat questionablebecause, by definition, all values assumed by X are supposed to lie withininterval (0, ). However, we see from Equation (9.75) that it is possible thatsome of the samples are greater than ^ . Intuitively, a better estimator might be^ ˆ X …n† ;…9:76†where X (n) is the nth-order statistic. As we will see, this would be the outcomefollowing the method of maximum likelihood, to be discussed in the nextsection.Since the method of moments requires only i , the moments of population X,the knowledge of its pdf is not necessary. This advantage is demonstrated inExample 9.13.Ex ample 9. 13. Problem: consider measuring the length r of an object with useof a sensing instrument. Owing to inherent inaccuracies in the instrument, whatis actually measured is X, as shown in Figure 9.3, where X 1 and X 2 areidentically and normally distributed with mean zero and unknown variance 2 . Determine a moment estimator ^ for ˆ r 2 on the basis of a sample of sizen from X.Answer: now, random variable X isX ˆ‰…r ‡ X 1 † 2 ‡ X 2 2 Š1=2 :…9:77†The pdf of X with unknown parameters and 2 can be found by usingtechniques developed in Chapter 5. It is, however, unnecessary here since someTLFeBOOK