  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).
