Exercise7.5 Continue Example7.5fortestingtwosimplehypotheses H0 : = 0 vs. H1 : = 1 with thedifferentlosses K0

Exercise7.5 Continue Example7.5fortestingtwosimplehypotheses H0 : θ = θ0 vs. H1 : θ = θ1 with thedifferentlosses K0 and K1 for erroneousrejectionandacceptanceofthenull.Assumeprior probabilities π0 and π1 = 1−π0 for H0 and H1 respectively.

1. DerivetheresultingBayesrule(test).

2. Interpret theBayestestfromthepreviousparagraphintermsofthefrequentistapproach.Isit LRT(MPtest)?Whatisthecriticalvalueforthecorrespondingteststatistic?

3. ShowthatanyLRTisadmissiblew.r.t. K0−K1 loss.

4. Let α and β be TypeIandTypeIIerrorsprobabilitiesrespectivelyfortheabovetest.Showthat if thecriticalvalueforLRTischosensuchthat K0α = K1β, thenitisalsoaminimaxtest.

5. Let Y1, ...,Y100 ∼ N (μ,25). Findtheminimaxtestfortesting H0 : μ = 0 vs. H1 : μ = 1 for K0 = 25 and K1 = 10.