# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 04Moment MatchingMoo K. Chung[email protected]September 14, 20041.Given a random sampleX1,· · ·, Xn, a linearestimator of parameterθis an estimator offormˆθ=nXi=1ciXi.Then it can be shown that¯Xis the MVUEfor population mean among all linear unbi-ased estimators.Proof.Casen= 2will be proved. The gen-eral statement follows inductively. Considerlinear estimatorsˆμ=c1X1+c2X2.To be unbiased,c1+c2= 1. To be most ef-ficient among all unbiased linear estimators,the variance has to be minimized. The vari-ance isVˆμ=c21VX1+c22VX2=£c21+ (1-c1)2/σ2The quadratic term in the bracket2c21-2c1+1is minimized whenc1= 1/2.2.Given a random sampleX1,· · ·, Xn, thek-th sample momentisMk=nj=1Xkj/n.Themoment estimatorsof population param-eters are obtained by matching the samplemoments to correspond population momentsand solving the resulting equations simulta-
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