Higher order influence functions and minimax estimation of nonlinear functionals

Details Higher order influence functions and minimax estimation of nonlinear functionals Higher order influence functions and minimax estimation of nonlinear functionals

2008-05-20

arxiv.org/abs/0805.3040v1

IMS Collections 2008, Vol. 2, 335-421

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/193940307000000527 the IMS Collections (http://www.imstat.org/publications/imscollec

Scientific paper

10.1214/193940307000000527

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).

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