Solution
= Solution
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.