One-step Estimator

Definition

Corrects first-order bias by adding the empirical mean of the estimated EIF to the plug-in $\psi(\hat P)$: $\hat\psi_{\text{os}}=\psi(\hat P)+\frac{1}{n}\sum_{i=1}^n\phi(O_i;\hat P).$ A single Newton step toward the efficient estimating equation. Under Cross-fitting (which avoids the Donsker condition) plus a nuisance convergence rate of $o_p(n^{-1/4})$, it is $\sqrt n$-consistent, asymptotically efficient, and double robust. The AIPW estimator for the ATE is the canonical one-step estimator.

Intuitive Understanding

“The prediction (plug-in) is biased → subtract its first-order bias using the IF.” Because the residual $R_2$ in the von Mises expansion is second-order, $\sqrt n$ is guaranteed even when the nuisance estimates converge somewhat slowly.

Related Concepts

• Influence Function · Efficient Influence Function · AIPW · TMLE (alternative correction) · Cross-fitting · Double Machine Learning

Key Papers

• Kennedy, “Semiparametric DR targeted DML: a review”, arXiv:2203.06469, 2022
• Pfanzagl (one-step/von Mises lineage); Tsiatis 2006