(i) Let $X$ be a random variable with density function $f(x ; \theta)$. Consider testing the simple null hypothesis $H_{0}: \theta=\theta_{0}$ against the simple alternative hypothesis $H_{1}: \theta=\theta_{1}$.

What is the form of the optimal size $\alpha$ classical hypothesis test?

Compare the form of the test with the Bayesian test based on the Bayes factor, and with the Bayes decision rule under the 0-1 loss function, under which a loss of 1 is incurred for an incorrect decision and a loss of 0 is incurred for a correct decision.

(ii) What does it mean to say that a family of densities $\{f(x ; \theta), \theta \in \Theta\}$ with real scalar parameter $\theta$ is of monotone likelihood ratio?

Suppose $X$ has a distribution from a family which is of monotone likelihood ratio with respect to a statistic $t(X)$ and that it is required to test $H_{0}: \theta \leqslant \theta_{0}$ against $H_{1}: \theta>\theta_{0}$.

State, without proof, a theorem which establishes the existence of a uniformly most powerful test and describe in detail the form of the test.

Let $X_{1}, \ldots, X_{n}$ be independent, identically distributed $U(0, \theta), \theta>0$. Find a uniformly most powerful size $\alpha$ test of $H_{0}: \theta \leqslant \theta_{0}$ against $H_{1}: \theta>\theta_{0}$, and find its power function. Show that we may construct a different, randomised, size $\alpha$ test with the same power function for $\theta \geqslant \theta_{0}$.