The test statistic is generated by the functional $T(F)=\int q_c(x)\,dF(x)$. At $F=P_0$, its influence function is
$$
\operatorname{IF}_{test}(x;T,P_0)
=q_c(x)-\mathbb E_{P_0}[q_c(X)].
$$
Because $|q_c(x)|\leq c$, this influence function is bounded. Multiplying $q_c$ by the constant inherited from the clipped log-likelihood ratio only rescales the same bounded expression.


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 3 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-223/3/e/solution