Statistical functionals and gross-error influence functions

Let $(\mathcal{X},\mathcal{A})$ be a measurable space, let $\mathcal{P}$ be a collection of probability measures on it, and let $T:\mathcal{P}\to {\mathbb{R}}^k$ be a single-valued map with finite $k\ge 1$. Such a map is a statistical functional. For $F\in \mathcal{P}$ and $x\in \mathcal{X}$, suppose the point-contamination path $F_{ϵ,x}=(1-ϵ)F+ϵ{\delta }_x$ belongs to $\mathcal{P}$ for all sufficiently small $ϵ\ge 0$ (Gross-error contamination path). The gross-error influence function at $(x;T,F)$ is

$\mathrm{IF}(x;T,F):={\lim }_{ϵ\downarrow 0}\frac{T(F_{ϵ,x})-T(F)}{ϵ}\in {\mathbb{R}}^k,$

when this finite right limit exists. Its domain may be a proper subset of $\mathcal{X}$.

This is a pointwise derivative along one point-contamination direction. Its definition supplies no remainder bound uniform in $x$ or in a general contaminating law $H$.

There are two other notions also called influence functions in semiparametric work. Pathwise differentiability and parameter influence curves defines pathwise differentiability through allowed differentiable-in-quadratic-mean paths and a continuous linear derivative on the score tangent space; this definition neither assumes nor asserts that the gross-error path is one of those paths. Asymptotically linear estimators and influence functions defines an estimator influence function through an empirical-average expansion with an $o_P(n^{-1\mathrm{/}2})$ remainder. Neither notion is identified with the pointwise gross-error derivative here. The pathwise supplier's Riesz-representation and choice-dependent conclusions are cited only as terminology context; no such conclusion or axiom is used in this definition.