Dr. Hackney STA Solutions pg 96

Dr. Hackney STA Solutions pg 96 - 6-10Solutions Manual for...

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Unformatted text preview: 6-10Solutions Manual for Statistical Inference6.40 a. BecauseX1,...,Xnis from a location scale family, by Theorem 3.5.6, we can writeXi=Zi+, whereZ1,...,Znis a random sample from the standard pdff(z). ThenT1(X1,...,Xn)T2(X1,...,Xn)=T1(Z1+,...,Zn+)T2(Z1+,...,Zn+)=T1(Z1,...,Zn)T2(Z1,...,Zn)=T1(Z1,...,Zn)T2(Z1,...,Zn).BecauseT1/T2is a function of onlyZ1,...,Zn, the distribution ofT1/T2does not dependonor; that is,T1/T2is an ancillary statistic.b.R(x1,...,xn) =x(n)-x(1). Becausea >0, max{ax1+b,...,axn+b}=ax(n)+bandmin{ax1+b,...,axn+b}=ax(1)+b. Thus,R(ax1+b,...,axn+b) = (ax(n)+b)-(ax(1)+b) =a(x(n)-x(1)) =aR(x1,...,xn). For the sample variance we haveS2(ax1+b,...,axn+b)=1n-1X((axi+b)-(ax+b))2=a21n-1X(xi-x)2=a2S2(x1,...,xn).Thus,S(ax1+b,...,axn+b) =aS(x1,...,xn). Therefore,RandSboth satisfy the abovecondition, andR/Sis ancillary by a).6.41 a. Measurement equivariance requires that the estimate ofbased onybe the same as theestimate ofbased onx; that is,T*(x1...
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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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