Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 27I / Solution

Interpret $\varphi (x)$ as the conditional rejection probability. The Type I error is $\alpha =E_0\varphi (X)$, the Type II error is $\beta =E_1[1-\varphi (X)]$, the size is $\alpha$, and the power is $1-\beta$. Their weighted sum isPointwise minimization over $0\le \varphi \le 1$ forces rejection where $p_1>c{p}_0$ and acceptance where $p_1<c{p}_0$, with arbitrary randomization on equality. Conversely those choices minimize every integrand. The equivalence holds almost everywhere, since changing a test on a common null set cannot change its errors. With both weights strictly positive, any simultaneous improvement of the two errors, strict in at least one, would lower this weighted sum. Every such positive-threshold likelihood ratio test is admissible for the two simple hypotheses.

For the specified densities the likelihood ratio isIt increases strictly on $(\theta ,1)$ from zero to $1/(1-\theta )$. For $0<c<1/(1-\theta )$, the likelihood-ratio test rejects when $x>\theta /[1-c(1-\theta {)}^2]$. Larger thresholds accept almost surely. Threshold zero rejects everywhere above $\theta$ and permits arbitrary rejection below $\theta$, where both weighted choices tie; values at individual endpoints do not affect errors. These describe all nonnegative-threshold likelihood ratio tests, including their limiting accept-all and reject-all cases.

The upper-tail test ${\varphi }_k$ has size $1-k^2$. If $\theta <k$, it is a positive-threshold likelihood ratio test and is admissible. If $\theta =k$, it rejects exactly the alternative's support, with power one; no test can retain power one with smaller size, so it remains admissible. If $\theta >k$, it still has power one but wastes rejection probability on $(k,\theta )$, where the alternative density vanishes. The test ${\varphi }_{\theta }$ has the same power and smaller size, soNevertheless ${\varphi }_k$ is a uniformly most powerful test at size $1-k^2$ against all $0<\theta <1$: for $\theta <k$, apply the Neyman-Pearson lemma; for $\theta \ge k$, its power is already the maximum one. It is also admissible against the composite alternative. Indeed choose any $0<\theta <k$. Strict monotonicity of its likelihood ratio makes ${\varphi }_k$ the unique positive-threshold minimizer up to null sets. A test dominating it for the composite problem would either improve the weighted simple risk at this $\theta$, impossible, or tie that risk and coincide almost everywhere. Since $p_0>0$ throughout $(0,1)$, such coincidence also precludes improvement at any other $\theta$.