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Unformatted text preview: 6-8Solutions Manual for Statistical Inference(ii) Ifis fixed,T=iXiis a complete sufficient statistic forby Theorem 6.2.25. Becauseis a scale parameter, ifZ1,...,Znis a random sample from a gamma(,1) distribution,thenX(i)/Thas the same distribution as (Z(i))/(iZi) =Z(i)/(iZi), and thisdistribution does not depend on. Thus,X(i)/Tis ancillary, and by Basus Theorem, itis independent ofT. We haveE(X(i)|T) = EX(i)TT T=TEX(i)TTindep.=TEX(i)Tpart (i)=TE(X(i))ET.Note, this expression is correct for each fixed value of (,), regardless whetherisknown or not.6.32 In the Formal Likelihood Principle, takeE1=E2=E. Then the conclusion is Ev(E,x1) =Ev(E,x2) ifL(|x1)/L(|x2) =c. Thus evidence is equal whenever the likelihood functions areequal, and this follows from Formal Sufficiency and Conditionality.6.33 a. For all sample points except (2,x*2) (but including (1,x*1)),T(j,xj) = (j,xj). Hence,g(T(j,xj)|)h(j,xj) =g((j,xj)|)1 =f*((j,xj)|)....
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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.
- Spring '12