Efficient Influence Function

Definition

Among the regular asymptotically linear (RAL) estimators of a (semi)parametric model, the IF with the smallest variance is the efficient influence function (EIF), and its variance equals the semiparametric efficiency bound (the supremum of the Cramér-Rao bounds over all parametric submodels). The EIF is the projection of an arbitrary IF onto the tangent space $\mathcal{T}$.

The EIF of the ATE ($\psi=E[Y(1)-Y(0)]$, under unconfoundedness): $\phi(O)=\mu_1(X)-\mu_0(X)-\psi+\frac{A\,(Y-\mu_1(X))}{e(X)}-\frac{(1-A)(Y-\mu_0(X))}{1-e(X)},$ i.e., the IF of the AIPW estimator.

Intuitive Understanding

The best achievable asymptotic precision under nonparametric assumptions. Using the EIF as an estimating equation (solving it to zero) yields an efficient and doubly robust estimator.

Related Concepts

• Influence Function · One-step Estimator · TMLE · AIPW
• Neyman-Orthogonal Score — the EIF is Neyman-orthogonal
• Double Machine Learning · Cross-fitting

Key Papers

• Tsiatis, Semiparametric Theory and Missing Data, Springer 2006 — tangent space and AIPW framework
• van der Vaart, Asymptotic Statistics, 1998, Ch.25 “Semiparametric Models”
• Bickel, Klaassen, Ritov & Wellner (BKRW), 1993 — the canonical reference in the field
• Kennedy review, arXiv:2203.06469, 2022