Past exam of the options II course of the University of Oxford / 2004 / O / O4

Let $f_0,f_1$ be densities or probability mass functions relative to a common measure, describing two simple hypotheses. A randomized hypothesis test is a measurable function $0\le \phi \le 1$, whose value is the conditional rejection probability. Fix $0<\alpha <1$. The Neyman-Pearson lemma states that a test ${\phi }_{\ast }$ with ${\mathbb{E}}_0{\phi }_{\ast }=\alpha$ is most powerful among tests with size at most $\alpha$ if it rejects where $f_1>c{f}_0$, accepts where $f_1<c{f}_0$, and randomizes as necessary on equality, for a threshold $c\ge 0$. Sets where $f_0=0<f_1$ are included in rejection. A likelihood-ratio quantile and boundary randomization supply such a test.

To prove optimality, let $\phi$ be any competitor with ${\mathbb{E}}_0\phi \le \alpha$. Pointwise, the threshold rule makesIntegrating and rearranging givesThus no competitor of the permitted size has larger statistical power, proving the lemma. Randomization permits exact size when discrete observations cannot supply it with a deterministic rejection region.

The likelihood ratio for the two binomial probabilities isstrictly increasing in $y$. By the Neyman-Pearson lemma, reject in the highest upper tail. Under the null,Hence reject exactly when $Y\ge 4$, with no randomization required. Its statistical power at the stated alternative is

For any fixed $p>1/4$ below 1, the ratio to the null isIt increases with $y$. Its ordering is therefore the same for every such alternative, while the size is determined only by the null. The very same rule, reject when $Y\ge 4$, is most powerful against each alternative and hence is a uniformly most powerful test. At $p=1$ its power is 1, also optimal.

More generally, for $0<p_0<1$, the binomial likelihood ratio against any $p>p_0$ increases with $y$. Choose an integer cutoff $c$ so thatand reject for $Y>c$, randomize with probabilitywhen $Y=c$, and accept below. This gives exact size $\alpha$ and is a uniformly most powerful test for every positive integer $n$. Exact prescribed size may require this boundary randomization; a deterministic cutoff alone need not attain it. For $p_0=0$, reject all $Y>0$ and reject at $Y=0$ with probability $\alpha$, which is likewise optimal. At $p_0=1$ the stated one-sided alternative is empty. The cutoff generally depends on $p_0,n,\alpha$; there is no claim that one fixed cutoff serves every null parameter.