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www.downloadslide.comCHAPTER9Properties of Point.Estimators and Methods.of Estimation.9.1 Introduction.9.2 Relative Efficiency.9.3 Consistency.9.4 Sufficiency.9.5 The Rao–Blackwell Theorem and Minimum-Variance Unbiased Estimation.9.6 The Method of Moments.9.7 The Method of Maximum Likelihood.9.8 Some Large-Sample Properties of Maximum-Likelihood Estimators (Optional).9.9 Summary.References and Further Readings9.1 Introduction.In Chapter 8, we presented some intuitive estimators for parameters often of interest.ˆ.in practical problems. An estimator θ for a target parameter θ is a function of the.random variables observed in a sample and therefore is itself a random variableConsequently, an estimator has a probability distribution, the sampling distribution.ˆ.of the estimator. We noted in Section 8.2 that, if E(θ) = θ, then the estimator has the.(sometimes) desirable property of being unbiasedIn this chapter, we undertake a more formal and detailed examination of some of the.mathematical properties of point estimators—particularly the notions of efficiency,.consistency, and sufficiency. We present a result, the Rao–Blackwell theorem, that.provides a link between sufficient statistics and unbiased estimators for parametersGenerally speaking, an unbiased estimator with small variance is or can be made to be.444.www.downloadslide.com.9.2Relative Efficiency445a function of a sufficient statistic. We also demonstrate a method that can sometimes.be used to find minimum-variance unbiased estimators for parameters of interest. We.then offer two other useful methods for deriving estimators: the method of moments.and the method of maximum likelihood. Some properties of estimators derived by.these methods are discussed.9.2 Relative Efficiency.It usually is possible to obtain more than one unbiased estimator for the same target.ˆ.ˆ.parameter θ . In Section 8.2 (Figure 8.3), we mentioned that if θ1 and θ2 denote two.unbiased estimators for the same parameter θ, we prefer to use the estimator with.ˆ.the smaller variance. That is, if both estimators are unbiased, θ1 is relatively more.ˆ.ˆ.ˆ.ˆ.ˆ2 if V (θ2 ) > V (θ1 ). In fact, we use the ratio V (θ2 )/V (θ1 ) to define the.efficient than θ.relative efficiency of two unbiased estimators.DEFINITION 9.1ˆ.ˆ.Given two unbiased estimators θ1 and θ2 of a parameter θ, with variances.ˆ.ˆ.ˆ.ˆ1 ) and V (θ2 ), respectively, then the efficiency of θ1 relative to θ2 , denoted.V (θ.ˆ.ˆ1 , θ2 ), is defined to be the ratio.eff (θ.ˆ.V (θ2 ).ˆ ˆ.eff (θ1 , θ2 ) =.ˆ.V (θ1 )ˆ.ˆ.ˆ.ˆ.If θ1 and θ2 are unbiased estimators for θ, the efficiency of θ1 relative to θ2 ,.ˆ.ˆ.ˆ.ˆ ˆ.eff (θ1 , θ2 ), is greater than 1 only if V (θ2 ) > V (θ1 ). In this case, θ1 is a better unbiased.ˆ ˆ.ˆ.ˆ.ˆ.estimator than θ2 . For example, if eff (θ1 , θ2 ) = 1.8, then V (θ2 ) = (1.8)V (θ1 ), and.ˆ.ˆ ˆ.ˆ.ˆ.θ1 is preferred to θ2 . Similarly, if eff (θ1 , θ2 ) is less than 1—say, .73—then V (θ2 ) =.ˆ.ˆ.ˆ.(.73)V (θ1 ), and θ2 is preferred to θ1 . Let us consider an example involving two.different estimators for a population mean. Suppose that we wish to estimate the.ˆ.mean of a normal population. Let θ1 be the sample median, the middle observation.when the sample measurements are ordered according to magnitude (n odd) or the.ˆ.average of the two middle observations (n even). Let θ2 be the sample mean. Although.proof is omitted, it can be shown that the variance of the sample median, for large.ˆ.n, is V (θ1 ) = (1.2533)2 (σ 2 /n). Then the efficiency of the sample median relative to.the sample mean is.ˆ.1.V (θ2 ).σ 2 /n.ˆ ˆ.eff (θ1 , θ2 ) =.=.= .6366=.ˆ1 ).(1.2533)2 σ 2 /n.(1.2533)2.V (θ.Thus, we see that the variance of the sample mean is approximately 64% of the.variance of the sample median. Therefore, we would prefer to use the sample mean.as the estimator for the population meanwww.downloadslide.com.446Chapter 9Properties of Point Estimators and Methods of EstimationE X A M PL E 9.1Let Y1 , Y2 , . . . , Yn denote
