Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 18H (Statistics)

A possibly randomized test is a function $\phi (X)\in [0,1]$, interpreted as the conditional probability of rejecting $H_0$. Its power function isIt has size at most $\alpha$ whenSuch a test is a uniformly most powerful test of size $\alpha$ if, for every other test $\psi$ of size at most $\alpha$,Thus one test maximizes power simultaneously at every parameter value in the alternative.

The likelihood ratio of the simple alternative to the simple null isThis is a strictly increasing function of $x$. By the Neyman-Pearson lemma, the most powerful test therefore rejects for $X>c$. The logistic distribution hasThe size condition ${\mathbb{P}}_0(X>c)=\alpha$ givesBecause the distribution is continuous, no boundary randomization is needed. The most powerful size-$\alpha$ test is therefore

For ${\theta }_1>{\theta }_0$, the likelihood ratio iswhich is increasing in $x$. Thus the logistic location family has a monotone likelihood ratio in $X$.

The upper-tail rejection probability is increasing in the location parameter, since $X=\theta +Z$ with $Z$ having the standard logistic distribution. Hence for the critical value from part (i),The test has size $\alpha$ for the composite null.

Now fix any alternative ${\theta }_1>0$. The Neyman-Pearson lemma applied to the simple hypotheses $\theta =0$ and $\theta ={\theta }_1$ says that this same upper-tail test is most powerful among all tests whose rejection probability at $0$ is at most $\alpha$. Every test of size at most $\alpha$ for the composite null satisfies that restriction. The upper-tail test is therefore most powerful at every ${\theta }_1>0$, so it is the uniformly most powerful upper-tail test for a logistic location.