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Robins et al, 2008, published a theory of higher order influence functionsfor inference in semi- and non-parametric models. This paper is a comprehensivemanuscript from which Robins et al, was drawn. The current paper includes manyresults and proofs that were not included in Robins et al due to spacelimitation. Particular results contained in the present paper that were notreported in Robins et al include the following. Given a set of functionals andtheir corresponding higher order influence functions, we show how to derive thehigher order influence function of their product. We apply this result toobtain higher order influence functions and associated estimators for the meanof a response Y subject to monotone missingness under missing at random. Theseresults also apply to estimating the causal effect of a time dependenttreatment on an outcome Y in the presence of time-varying confounding. Finally,we include an appendix that contains proofs for all theorems that were statedwithout proof in Robins et al, 2008. The initial part of the paper is closelyrelated to Robins et al, the latter parts differ.

Technical Report: Higher Order Influence Functions and Minimax Estimation of Nonlinear Functionals