Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 18H (Statistics)

Write $X_i=\theta +U_i$, where the $U_i$ are independent $\mathrm{Uniform}(0,1)$ variables. Convolution gives the sum of two independent uniform variables:Its plot is a triangle with vertices

For Test 1, the rejection probability is increasing in $\theta$, so its size over $\theta \le 0$ is attained at $\theta =0$ and equalsTest 2 is also monotone under shifts in $\theta$, so its size is ${\mathbb{P}}_0(U_1+U_2>C)$. For $1\le C\le 2$, the triangular upper tail isEquating this to $0.05$ gives

The power function of a statistical test for Test 1 is

Put $a=1/\sqrt{10}$. Test 2 rejects whenUsing the two branches of the triangular distribution givesBoth functions equal $0.05$ at the least favourable null value $\theta =0$.

The two tests have the same size, but their powers cross. At $\theta =0.1$,At $\theta =0.5$,Thus Test 2 is a same-size test that beats Test 1 at one alternative, while Test 1 beats Test 2 at another. By the crossing-power obstruction to a uniformly most powerful test,