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Dr. Hackney STA Solutions pg 135

Dr. Hackney STA Solutions pg 135 - 8-14Solutions Manual for...

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Unformatted text preview: 8-14Solutions Manual for Statistical Inferenceb. Forθ≤k-1,β(θ) = 0. Fork-1< θ≤0,β(θ) =Zθ+1kn(1-(y1-θ))n-1dy1= (1-k+θ)n.For 0< θ≤k,β(θ) =Zθ+1kn(1-(y1-θ))n-1dy1+ZkθZθ+11n(n-1)(yn-y1)n-2dyndy1=α+ 1-(1-θ)n.And fork < θ,β(θ) = 1.c. (Y1,Yn) are sufficient statistics. So we can attempt to find a UMP test using Corollary 8.3.13and the joint pdff(y1,yn|θ) in part (a). For 0< θ <1, the ratio of pdfs isf(y1,yn|θ)f(y1,yn|0)=if 0< y1≤θ,y1< yn<11ifθ < y1< yn<1∞if 1≤yn< θ+ 1,θ < y1< yn.For 1≤θ, the ratio of pdfs isf(y1,yn|θ)f(y1,yn|0)=ify1< yn<1∞ifθ < y1< yn< θ+ 1.For 0< θ < k, usek= 1. The given test always rejects iff(y1,yn|θ)/f(y1,yn|0)>1 andalways accepts iff(y1,yn|θ)/f(y1,yn|0)<1. Forθ≥k, usek= 0. The given test alwaysrejects iff(y1,yn|θ)/f(y1,yn|0)>0 and always accepts iff(y1,yn|θ)/f(y1,yn|0)<0. Thusthe given test is UMP by Corollary 8.3.13....
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