Dr. Hackney STA Solutions pg 76

Dr. Hackney STA Solutions pg 76 - 5-10Solutions Manual for...

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Unformatted text preview: 5-10Solutions Manual for Statistical InferenceNow letW=Z/Y,Q=Y. ThenY=Q,Z=WQ, and|J|=q. ThereforefW,Q(w,q) =n(n-1)n(q-wq)n-2q=n(n-1)n(1-w)n-2qn-1,< w <1,< q < .The joint pdf factors into functions ofwandq, and, hence,WandQare independent.5.25 The joint pdf ofX(1),...,X(n)isf(u1,...,un) =n!ananua-11ua-1n,< u1<< un< .Make the one-to-one transformation toY1=X(1)/X(2),...,Yn-1=X(n-1)/X(n),Yn=X(n).The Jacobian isJ=y2y23yn-1n. So the joint pdf ofY1,...,Ynisf(y1,...,yn)=n!anan(y1yn)a-1(y2yn)a-1(yn)a-1(y2y23yn-1n)=n!ananya-11y2a-12yna-1n,< yi<1;i= 1,...,n-1,< yn< .We see thatf(y1,...,yn) factors soY1,...,Ynare mutually independent. To get the pdf ofY1, integrate out the other variables and obtain thatfY1(y1) =c1ya-11, 0< y1<1, for someconstantc1. To have this pdf integrate to 1, it must be thatc1=a. ThusfY1(y1) =aya-11,< y1<1. Similarly, fori= 2,...,n-1, we obtainfYi(yi) =iayia-1i,< yi<1.FromTheorem 5.4.4, the pdf ofYnisfYn(yn) =na...
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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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