Minimax Estimation of Nonlinear Functionals with Higher Order Influence Functions: Results and Applications

Professor Robins describes a novel approach to point and interval estimation of nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions. Higher order influence functions are higher order U-statistics. The approach applies equally to both n‾√ and non-n‾√ problems. It reproduces results previously obtained by the modern theory of non-parametric inference, produces many new n‾√ and non-n‾√ results, and opens up the ability to perform non-n‾√ inference in complex high dimensional models, such as models for the average causal effects of not only time independent treatments but also of time varying treatments in the presence of time varying confounding variables and informative censoring. The estimators are always rate minimax in n‾√ problems and often in non-n‾√ problems as well. The approach can also be used to derive novel nonparametric tests of unconditional and conditional independence with good power against complex alternatives.