Higher order influence functions and minimax estimation of nonlinear functionals

TL;DR

This paper develops a comprehensive theory for estimating nonlinear functionals using higher order influence functions, enabling optimal inference in complex models where traditional methods fail to achieve root-n rates.

Contribution

It introduces a novel framework extending semiparametric theory with higher order influence functions, allowing for optimal non-root-n estimation in high-dimensional settings.

Findings

Provides rate-optimal estimators for complex functionals

Extends semiparametric theory to higher order influence functions

Demonstrates multi-robustness property of influence functions

Abstract

We present a theory of point and interval estimation for nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions (Robins (2004), Section 9; Li et al. (2004), Tchetgen et al. (2006), Robins et al. (2007)). Higher order influence functions are higher order U-statistics. Our theory extends the first order semiparametric theory of Bickel et al. (1993) and van der Vaart (1991) by incorporating the theory of higher order scores considered by Pfanzagl (1990), Small and McLeish (1994) and Lindsay and Waterman (1996). The theory reproduces many previous results, produces new non-$\sqrt{n}$ results, and opens up the ability to perform optimal non-$\sqrt{n}$ inference in complex high dimensional models. We present novel rate-optimal point and interval estimators for various functionals of central importance to biostatistics in settings in which estimation at the expected $\sqrt{n}$ rate is not possible, owing to the curse of dimensionality. We also show that our higher order influence functions have a multi-robustness property that extends the double robustness property of first order influence functions described by Robins and Rotnitzky (2001) and van der Laan and Robins (2003).