 Solution

Let $f_0,f_1$ be densities or probability mass functions relative to a common measure, describing two simple hypotheses. A randomized hypothesis test is a measurable function $0\le \phi \le 1$, whose value is the conditional rejection probability. Fix $0<\alpha <1$. The Neyman-Pearson lemma states that a test ${\phi }_{\ast }$ with ${\mathbb{E}}_0{\phi }_{\ast }=\alpha$ is most powerful among tests with size at most $\alpha$ if it rejects where $f_1>c{f}_0$, accepts where $f_1<c{f}_0$, and randomizes as necessary on equality, for a threshold $c\ge 0$. Sets where $f_0=0<f_1$ are included in rejection. A likelihood-ratio quantile and boundary randomization supply such a test.

To prove optimality, let $\phi$ be any competitor with ${\mathbb{E}}_0\phi \le \alpha$. Pointwise, the threshold rule makesIntegrating and rearranging givesThus no competitor of the permitted size has larger statistical power, proving the lemma. Randomization permits exact size when discrete observations cannot supply it with a deterministic rejection region.