= Influence function
{title2=$\operatorname{IF}(x;T,F)$}
{wiki}
= Influence curve
{synonym}
An <influence function> asks how much a population answer moves when an infinitesimal fraction of its <probability distribution> is relocated to a specified point. It distinguishes the unbounded response of the <mean> from the bounded response of the <median> and the <Huber location estimator>. The <normal-location influence curves> and <finite contamination bias of normal location> below make that difference quantitative.
Let $T$ be a real-valued <statistical functional>, $F$ a reference <probability measure>, and $\delta_x$ the <Dirac measure> concentrated at $x$. Along the <epsilon-contamination neighborhood> path $F_{\varepsilon,x}=(1-\varepsilon)F+\varepsilon\delta_x$, define the right <derivative>
$$
\operatorname{IF}(x;T,F)=\lim_{\varepsilon\downarrow0}\frac{T(F_{\varepsilon,x})-T(F)}{\varepsilon},\qquad
T(F_{\varepsilon,x})=T(F)+\varepsilon\operatorname{IF}(x;T,F)+o(\varepsilon).
$$
The name <influence curve> refers to the same object. The <derivative> is in the signed direction $\delta_x-F$, so the calculation removes baseline mass as well as adding a <Dirac measure>. The definition is a <limit> for each fixed $x$; it does not automatically give a remainder uniform over arbitrarily distant $x$.<The Influence Curve and Its Role in Robust Estimation by Frank R. Hampel>[Frank R. Hampel, The Influence Curve and Its Role in Robust Estimation, p. 383, §2: pointwise right-limit definition along mixtures with a point mass]{ref}
The <statistical functional> must be defined on these contaminated <probability measures>, and the right <derivative> must exist and be finite. For a <mean>, a finite first absolute <moment> suffices. For a <median>, require a unique baseline <median> and a <probability density function> continuous and strictly positive there. For a smooth <M-estimator>, require an isolated population root, justified differentiation of its <expected value>, and a nonzero derivative with respect to the fitted parameter. These are specific conditions for the formulas below. Existence of the pointwise <influence function> alone proves neither an <asymptotic linear representation> nor <pathwise differentiability of a statistical functional>; each needs control of its own remainder and admissible <statistical paths>.
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