Higher order Inuence Functions and Minimax Estimation of Nonlinear Functionals

We present a theory of point and interval estimation for nonlinear functionals in parametric, semi-, and non-parametric models based on higher order in‡uence functions (Robins [18], Sec. 9; Li et al. [10], Tchetgen et al, [23], Robins et al, [20]). Higher order in‡uence functions are higher order U-statistics. Our theory extends the …rst order semiparametric theory of Bickel et al. [3] and van der Vaart [29] by incorporating the theory of higher order scores considered by Pfanzagl [13], Small and McLeish [22] and Lindsay and Waterman [9]. The theory reproduces many previous results, produces new nonp n results, and opens up the ability to perform optimal nonp 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 p n rate is not possible, owing to the curse of dimensionality. We also show that our higher order in‡uence functions have a multi-robustness property that extends the double robustness property of …rst order in‡uence functions described by Robins and Rotnitzky [19] and van der Laan and Robins [26]. 1 Introduction Over the past 3 years, we have developed a theory of point and interval estimation for nonlinear functionals (F ) in parametric, semi-, and non-parametric models based on higher order likelihood scores and in‡uence functions that applies equally to both p n and nonp n problems (Robins 2004, Sec. 9; Li et al, 2006, Tchetgen et al, 2006, Robins et al, 2007). The theory reproduces results previously obtained by the modern theory of non-parametric inference, produces many new nonp n results, and most importantly opens up the ability to perform nonp n inference in complex high dimensional models, such as models for the estimation of the causal e¤ect of time varying treatments in the presence of time varying confounding and informative censoring. See Tchetgen et al. (2007) for examples of the latter. Higher order in‡uence functions are higher order U-statistics. Our theory extends the …rst order semiparametric theory of Bickel et al. (1993) and van