5.(12pts) Suppose that we are given a random sampleX1,X2,···,Xnfrom thep.d.f.f(x) =θxθ−1, x∈(0,1),whereθ >0 is an unknown parameter. The null hypothesisH0:θ= 1 is to be tested against thealternativeH1:θ >1.(a) (4pts) Show that the family of uniformly most powerful tests have the rejection region of the form∏ni=1Xi> c0for some constantc0.(b) (8pts) LetX∼U(0,1) andY=−logX, show thatYfollows exponential distribution withλ= 1, i.e.,Y∼exp(1). Use this result to find a uniformly most powerful test for aboveH0:θ= 1vs.H1:θ >1. (i.e., determining the exact value of that constantc0in part (a) in terms of aknown distribution’s quartile using exact distribution of the test statistics) at significance levelα= 0.05.
6.(12pts) Suppose that we are given a random sampleX1,X2,···,XnfromN(0, σ2), whereσ >0 isan unknown parameter.(a) (6pts) Find the most powerful test for: the null hypothesisH0:σ=σ0is to be tested against thealternativeH1:σ=σ1at significance levelα= 0.05, whereσ1> σ0. The values ofσ0andσ1arefixed.(b) (6pts) Derive a likelihood ratio test ofH0:σ=σ0verseH1:σ6=σ0at significance levelα= 0.05.Provide the test statistic and rejection region as simple as possible and explain how to carry outabove hypothesis testH0:σ=σ0verseH1:σ6=σ0.
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