Cross-fitting and fast remainder rates for semiparametric estimation

There are many interesting and widely used estimators of a functional with ?nite semi-parametric variance bound that depend on nonparametric estimators of nuisance func-tions. We use cross-?tting to construct such estimators with fast remainder rates. We give cross-?t doubly robust estimators that use separate subsamples to estimate di?erent nuisance functions. We show that a cross-?t doubly robust spline regression estimator of the expected conditional covariance is semiparametric e?cient 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-?t plug-in estimator shares some of these properties but has a remainder term that is larger than the cross-?t doubly robust estimator. As speci?c examples we consider the expected conditional covariance, mean with randomly missing data, and a weighted average derivative.