 Solution

The power of a test at a parameter value is its probability of rejecting the null there. Its size is the supremum of this probability over the null parameter set. The Neyman-Pearson lemma states that for simple null and alternative densities $f_0,f_1$, a test rejecting where $f_1>c{f}_0$, accepting where $f_1<c{f}_0$, and randomizing as needed on equality to attain size $\alpha$, has greatest power among tests of size at most $\alpha$.

The specified function is nonnegative. Its integral isFor independent observations write $S={\sum }_i{X}_i$. The likelihood ratio of the smaller alternative parameter to the null isIt is strictly increasing in $S$. Thus the most powerful level-$\alpha$ test rejects forThe continuous distribution permits this size without randomization for $0<\alpha <1$. Its threshold depends only on the null and the desired size, not on which ${\theta }_1<{\theta }_0$ was chosen. The same test is therefore uniformly most powerful against the entire one-sided alternative $\theta <{\theta }_0$.