 Influence function of a quantile

Let $q$ be the unique $p$th quantile, with $0<p<1$, and suppose the density $f$ is continuous and positive at $q$. Define quantiles of contaminated distributions using the generalized inverse. For $z\ne q$, differentiating $(1-\epsilon )F(q_{\epsilon })+\epsilon 1_{z\le q_{\epsilon }}=p$ givesAt $z=q$, the generalized-inverse quantile stays exactly $q$ under this contamination, so its derivative is zero. The step formula thus holds $F$-almost everywhere and determines the asymptotic variance $p(1-p)/(nf(q{)}^2)$. Its gross-error sensitivity is $\max (p,1-p)/f(q)$, but its local-shift sensitivity is infinite because of the jump.