Gross-error sensitivity of a functional

For a statistical functional $T$ at a law $F$, let $D_{T,F}:=\{x\in \mathcal{X}:\mathrm{IF}(x;T,F)\text{ exists as a finite vector}\}$. When $D_{T,F}\mathrm{\ne }\mathrm{\varnothing }$, its gross-error sensitivity is the extended nonnegative number

${\gamma }^{\ast }(T,F):={\sup }_{x\in D_{T,F}}\mathrm{\parallel }\mathrm{IF}(x;T,F){\mathrm{\parallel }}_2\mathrm{.}$

For a real-valued functional the norm is absolute value. This supremum only ranges over directions where the influence function exists; it makes no claim that the derivative exists in every direction.

Finite gross-error sensitivity is a local pointwise stability bound. It does not, without a uniform remainder hypothesis, bound the maximum bias over an entire contamination neighborhood.