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

For a location M-functional defined byits influence function isprovided the denominator is nonzero. A bounded score therefore gives a bounded influence function.

The Huber location estimator usesaway from the two corners. At $F=N(\theta ,1)$, symmetry gives $T(F)=\theta$, and for $Z\sim N(0,1)$,Hencewhich is bounded in $x$.

For a randomized test $\phi (X_1,\dots ,X_n)\in [0,1]$, level $\alpha$ over the null neighborhood meansA maximin test maximizes its worst-case poweramong all tests satisfying that robust level constraint.

For $P_0=N(-{\theta }_0,1)$ and $P_1=N({\theta }_0,1)$, the nominal log-likelihood ratio for one observation is $2{\theta }_{0}x$. Under sufficiently small epsilon-contamination neighborhoods, the least-favourable pair clips this likelihood ratio between two constants. Symmetry turns its log into a positive multiple of the winsorized scoreThe robust form of the Neyman-Pearson lemma therefore rejects for large values ofwith boundary randomization and threshold chosen so that the worst-case null rejection probability is $\alpha$. Extreme observations contribute only $\pm c$, preventing a few contaminants from dominating the test.

The test statistic is generated by the functional $T(F)=\int q_c(x)dF(x)$. At $F=P_0$, its influence function isBecause $|q_c(x)|\le c$, this influence function is bounded. Multiplying $q_c$ by the constant inherited from the clipped log-likelihood ratio only rescales the same bounded expression.