Exercise4.7 Let Y1, ...,Yn be arandomsamplefromadistributionwiththedensityfunction f (y) = y e y2 2 , y

Exercise4.7 Let Y1, ...,Yn be arandomsamplefromadistributionwiththedensityfunction fθ (y) =

y

θ

e− y2 2θ , y ≥ 0, θ > 0 1. Find theMPtestatlevel α for testingtwosimplehypotheses H0 : θ = θ0 vs. H1 : θ = θ1, where

θ1 > θ0.

2. Is thereaUMPtestatlevel α for testingtheone-sidedcompositehypotheses H0 : θ ≤ θ0 vs.

H1 : θ > θ0? Whatisitspowerfunction?

(Hint: showthat Y2 i ∼ exp(1/(2θ)) and, therefore, Σni

=1Y2 i ∼ θ χ2 2n.)