Cross-fitting and fast remainder rates for semiparametric estimation

There are many interesting and widely used estimators of a functional with finite semi-parametric variance bound that depend on nonparametric estimators of nuisance func-tions. We use cross-fitting to construct such estimators with fast remainder rates. We give cross-fit doubly robust estimators that use separate subsamples to estimate different nuisance functions. We show that a cross-fit doubly robust spline regression estimator of the expected conditional covariance is semiparametric efficient under minimal conditions. Corresponding estimators of other average linear functionals of a conditional expectation are shown to have the fastest known remainder rates under certain smoothness conditions. The cross-fit plug-in estimator shares some of these properties but has a remainder term that is larger than the cross-fit doubly robust estimator. As specific examples we consider the expected conditional covariance, mean with randomly missing data, and a weighted average derivative.