= Influence function of a quantile
{title2=$\operatorname{IF}(z;q_p,F)$}
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 $(1-\varepsilon)F(q_\varepsilon)+\varepsilon\mathbf1_{z\le q_\varepsilon}=p$ gives
$$
\operatorname{IF}(z;q_p,F)=\frac{p-\mathbf1_{z\le q}}{f(q)}.
$$
At $z=q$, 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.
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