Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 223 / 2 / a / Solution

Put $q_s=F^{-1}(s)$ and $k=-F^{-1}(\alpha )>0$. Symmetry gives $F^{-1}(1-\alpha )=k$. Differentiating the trimmed mean functional along $F_t=(1-t)F+t{\Delta }_x$ givesUnder the substitution $y=q_s$, symmetry implieswhileEvaluating the last integral in the three regions $x<-k$, $|x|\le k$, and $x>k$ yieldsThis is the influence function of a trimmed mean, namely the Huber score with clipping parameter $k$, divided by $1-2\alpha$.