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 ofλ. Because E (∑iXi) =nλ, we musthave E (nX(1)|∑iXi) =∑iXi/nby completeness. Of course, any function of∑iXithatis an unbiased estimator ofλis the best unbiased estimator ofλ. Thus, we know directlythat because E(∑iXi) =nλ,∑iXi/nis the best unbiased estimator ofλ.c. From part (a),ˆλ= 601.2 and from part (b)ˆλ= 128.8. Maybe the exponential model is nota good assumption.7.50 a. E(a¯X+ (1-a)cS) =aE¯X+ (1-a)E(cS) =aθ+ (1-a)θ=θ. Soa¯X+ (1-a)cSis anunbiased estimator ofθ.b. Because¯XandS2are independent for this normal model, Var(a¯X+(1-a)cS) =a2V1+(1-a)2V2, whereV1= Var¯X=θ2/nandV2= Var(cS) =c2ES2-θ2=c2θ2-θ2= (c2-1)θ2.Use calculus to show that this quadratic function ofais minimized ata=V2V1+V2=(c2-1)θ2((1/n) +c2-1)θ2=(c2-1)((1/n) +c2-1)....
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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