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

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.