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VOL. 2 | 2008 Higher order influence functions and minimax estimation of nonlinear functionals
James Robins, Lingling Li, Eric Tchetgen, Aad van der Vaart

Editor(s) Deborah Nolan, Terry Speed

## 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).

## Information

Published: 1 January 2008
First available in Project Euclid: 7 April 2008

zbMATH: 05556804
MathSciNet: MR2459958

Digital Object Identifier: 10.1214/193940307000000527

Subjects:
Primary: 62C20 , 62G05

Keywords: influence functions , minimax , nonparametric , robust inference , U-statistics

Rights: Copyright © 2008, Institute of Mathematical Statistics

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Vol. 2 • 1 January 2008