Papers › Double Cross-fit Doubly Robust Estimators: Beyond Series Regression
Double Cross-fit Doubly Robust Estimators: Beyond Series Regression
Alec McClean, Sivaraman Balakrishnan, Edward H. Kennedy, Larry Wasserman
Doubly robust estimators with cross-fitting have gained popularity in causal inference due to their favorable structure-agnostic error guarantees. However, when additional structure, such as H\"{o}lder smoothness, is available then more accurate "double cross-fit doubly robust" (DCDR) estimators can be constructed by splitting the training data and undersmoothing nuisance function estimators on independent samples. We study a DCDR estimator of the Expected Conditional Covariance, a functional of interest in causal inference and conditional independence testing. We first provide a structure-agnostic error analysis for the DCDR estimator with no assumptions on the nuisance functions or their estimators. Then, assuming the nuisance functions are H\"{o}lder smooth, but without assuming knowledge of the true smoothness level or the covariate density, we establish that DCDR estimators with several linear smoothers are √(n)-consistent and asymptotically normal under minimal conditions and achieve fast convergence rates in the non-√(n) regime. When the covariate density and smoothnesses are known, we propose a minimax rate-optimal DCDR estimator based on undersmoothed kernel regression. Moreover, we show an undersmoothed DCDR estimator satisfies a slower-than-√(n) central limit theorem, and that inference is possible even in the non-√(n) regime. Finally, we support our theoretical results with simulations, providing intuition for double cross-fitting and undersmoothing, demonstrating where our estimator achieves √(n)-consistency while the usual "single cross-fit" estimator fails, and illustrating asymptotic normality for the undersmoothed DCDR estimator.
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