Past exam of the options II course of the University of Oxford / 2010 / O / O5

The significance level of a test is a bound $\alpha$ on its probability of rejecting a true null; for a composite null the bound is uniform over its parameter values. Its power at an alternative parameter is its rejection probability under that parameter.

The Neyman-Pearson lemma concerns a simple null density $f_0$ and a simple alternative density $f_1$ for the whole sample. Among tests of size at most $\alpha$, a most powerful test rejects where $f_1>k{f}_0$, accepts where $f_1<k{f}_0$, and if necessary randomizes on equality so that its size is $\alpha$. Let its rejection probability function be ${\varphi }_{\ast }$, with $0\le {\varphi }_{\ast }\le 1$, and let $\varphi$ be any competing test of size at most $\alpha$.

Pointwise,on the rejection side both factors are nonnegative, and on the acceptance side both are nonpositive. Integrating givesThus the likelihood-ratio test has at least as much power as any competitor. Even when the sample has a continuous density, the likelihood ratio can have an atom, so boundary randomization may still be necessary in the general statement. For the exponential test below the statistic has a continuous distribution and no randomization is needed.

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.