Cross-Fitting and Fast Remainder Rates for Semiparametric Estimation

TL;DR

This paper introduces cross-fitting techniques for semiparametric estimators, achieving faster remainder rates and efficiency in estimating functionals of conditional expectations, with applications to various statistical problems.

Contribution

It develops novel cross-fit doubly robust estimators with improved remainder rates and efficiency for semiparametric functionals, especially using spline regression.

Findings

Cross-fit doubly robust estimators achieve semiparametric efficiency.

These estimators have the fastest known remainder rates under certain conditions.

The cross-fit plug-in estimator has nearly the fastest rate but converges slower.

Abstract

There are many interesting and widely used estimators of a functional with finite semiparametric variance bound that depend on nonparametric estimators of nuisance functions. We use cross-fitting (i.e. sample splitting) to construct novel estimators with fast remainder rates. We give cross-fit doubly robust estimators that use separate subsamples to estimate different nuisance functions. We obtain general, precise results for regression spline estimation of average linear functionals of conditional expectations with a finite semiparametric variance bound. We show that a cross-fit doubly robust spline regression estimator of the expected conditional covariance is semiparametric efficient under minimal conditions. Cross-fit doubly robust estimators of other average linear functionals of a conditional expectation are shown to have the fastest known remainder rates for the Haar basis or under certain smoothness conditions. Surprisingly, the cross-fit plug-in estimator also has nearly the fastest known remainder rate, but the remainder converges to zero slower 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.