 Solution

For a fixed alternative $0<\lambda <{\lambda }_0$, the joint likelihood ratio isIt is strictly increasing in $S={\sum }_i{x}_i$. By the Neyman-Pearson lemma, the most powerful level-$\alpha$ rejection region is therefore $S\ge c_{\alpha }$. Under the null, $S\sim \mathrm{Gamma}(n,{\lambda }_0)$ with the second parameter interpreted as a rate. Its continuous distribution givesThis threshold depends on the null and $\alpha$, but not on the particular smaller alternative $\lambda$. Thus the same test is most powerful for every $0<\lambda <{\lambda }_0$, making it uniformly most powerful.

Under a true alternative $\lambda$, $S\sim \mathrm{Gamma}(n,\lambda )$, so its attained power isFor each $\lambda <{\lambda }_0$, the Neyman-Pearson lemma bounds the power of every other level-$\alpha$ test by this value. Hence the displayed function is not only an upper bound but the attainable power envelope on the specified alternative.