Influence function ($\mathrm{IF}(x;T,F)$)

The influence function is the derivative at zero contamination of a statistical functional $T((1-\epsilon )F+\epsilon {\delta }_x)$.

Gross-error sensitivity is the supremum of the absolute influence function over contamination points.

At a distribution with positive density $f$ at its median $m$, the median influence function is $\mathrm{sgn}(x-m)/(2f(m))$ away from $m$.