Dr. Hackney STA Solutions pg 117

Dr. Hackney STA Solutions pg 117 - 7-20Solutions Manual for...

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Unformatted text preview: 7-20Solutions Manual for Statistical Inferenceb. BecausefX(x) is in the exponential family,iXiis a complete sufficient statistic andE (nX(1)|iXi) is the best unbiased estimator of. Because E (iXi) =n, we musthave E (nX(1)|iXi) =iXi/nby completeness. Of course, any function ofiXithatis an unbiased estimator ofis the best unbiased estimator of. Thus, we know directlythat because E(iXi) =n,iXi/nis the best unbiased estimator of.c. From part (a),= 601.2 and from part (b)= 128.8. Maybe the exponential model is nota good assumption.7.50 a. E(aX+ (1-a)cS) =aEX+ (1-a)E(cS) =a+ (1-a)=. SoaX+ (1-a)cSis anunbiased estimator of.b. BecauseXandS2are independent for this normal model, Var(aX+(1-a)cS) =a2V1+(1-a)2V2, whereV1= VarX=2/nandV2= Var(cS) =c2ES2-2=c22-2= (c2-1)2.Use calculus to show that this quadratic function ofais minimized ata=V2V1+V2=(c2-1)2((1/n) +c2-1)2=(c2-1)((1/n) +c2-1)....
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