Uniformly most powerful test

A level-$\alpha$ test is uniformly most powerful when its power is at least that of every other level-$\alpha$ test at every parameter value in the alternative.

For a Poisson sample tested against mean ${\mu }_0=1$, the exact uniformly most powerful one-sided test thresholds the sample sum and randomizes on its boundary if necessary. A normal approximation gives critical sum $n+z_{1-\alpha }\sqrt{n}$. Equating its standardized alternative threshold to $-z_{1-\beta }$ gives the displayed design for approximate power $1-\beta$ at mean ${\mu }_1>1$. At size $0.05$ and target power $0.9$, the two quantiles are $1.645$ and $1.282$. Rounding up supplies an integer design; exact Poisson power still requires discrete evaluation.

If two tests have the same size and each has strictly greater power than the other at some alternative, neither test is uniformly most powerful.