Statistics | Part IB, 2004

Defining carefully the terminology that you use, state and prove the NeymanPearson Lemma.

Let XX be a single observation from the distribution with density function

f(xθ)=12exθ,<x<f(x \mid \theta)=\frac{1}{2} e^{-|x-\theta|}, \quad-\infty<x<\infty

for an unknown real parameter θ\theta. Find the best test of size α,0<α<1\alpha, 0<\alpha<1, of the hypothesis H0:θ=θ0H_{0}: \theta=\theta_{0} against H1:θ=θ1H_{1}: \theta=\theta_{1}, where θ1>θ0\theta_{1}>\theta_{0}.

When α=0.05\alpha=0.05, for which values of θ0\theta_{0} and θ1\theta_{1} will the power of the best test be at least 0.950.95 ?

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