Asymptotically linear estimators and influence functions

Fix a law $P$ and an iid sample $X_1,\dots ,X_n\sim P$. For a measurable vector function $\phi :\mathcal{X}\to {\mathbb{R}}^k$, write $P\phi =E_P[\phi (X)]$ and $P_n\phi =n^{-1}{\sum }_{i=1}^n\phi (X_i)$, where $P_n=n^{-1}{\sum }_{i=1}^n{\delta }_{X_i}$ is the empirical measure. For a vector remainder $R_n$, $R_n=o_P(n^{-1\mathrm{/}2})$ means $\sqrt{n}{R}_n\to 0$ in $P$-probability.

A measurable estimator sequence ${\widehat{\psi }}_n$ of $\psi (P)\in {\mathbb{R}}^k$ is asymptotically linear at $P$ with estimator influence function $\phi$ if

${\widehat{\psi }}_n-\psi (P)=(P_n-P)\phi +o_P(n^{-1\mathrm{/}2}),$

where $\phi \in L_0^2(P{)}^k$: $E_P\phi =0$ and $E_P\mathrm{\parallel }\phi {\mathrm{\parallel }}^2<\mathrm{\infty }$. The space $L_0^2(P)$ and its almost-sure equivalence convention are as in Mean-zero square-integrable functions. This definition alone does not assert regularity under local alternatives, pathwise differentiability of $\psi$, or asymptotic normality.

The term “influence function” here describes an estimator expansion. The neighboring pathwise-differentiability definition describes a functional's derivative along model paths. The two notions are distinct, though they can coincide for a particular estimator and target.