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, and derive a series of increasingly powerful results with progressively stronger assumptions. 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 semiparametric efficient under minimal conditions and achieve fast convergence rates in the non-$\sqrt{n}$ 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-$\sqrt{n}$ central limit theorem, and that inference is possible even in the non-$\sqrt{n}$ regime. Finally, we support our theoretical results with simulations, providing intuition for double cross-fitting and undersmoothing, demonstrating where our estimator achieves semiparametric efficiency while the usual "single cross-fit" estimator fails, and illustrating asymptotic normality for the undersmoothed DCDR estimator.

翻译：双重交叉拟合的双稳健估计量因其在与结构无关的误差保证方面的优势，在因果推断中日益受到关注。然而，当存在额外结构（如 Hölder 光滑性）时，可以通过分割训练数据并在独立样本上对辅助函数估计量进行欠平滑处理，构建更精确的“双重交叉拟合双稳健”（DCDR）估计量。本文研究因果推断与条件独立性检验中感兴趣的泛函——期望条件协方差的 DCDR 估计量，并在逐步增强的假设下推导出一系列渐近有力的结论。首先，我们在不对辅助函数或其估计量作任何假设的情况下，给出 DCDR 估计量的结构无关误差分析。其次，在假设辅助函数满足 Hölder 光滑性但未知真实光滑程度及协变量密度时，我们证明基于若干线性平滑器的 DCDR 估计量在最小条件下达到半参数有效性，并在非 $\sqrt{n}$ 收敛速率下实现快速收敛。当协变量密度与光滑性已知时，我们提出基于欠平滑核回归的极小极大最优 DCDR 估计量。进一步，我们证明欠平滑 DCDR 估计量满足慢于 $\sqrt{n}$ 的中心极限定理，且即使在非 $\sqrt{n}$ 收敛场景下仍可进行统计推断。最后，我们通过模拟实验支持理论结果：阐明双重交叉拟合与欠平滑的直观原理，展示所提估计量在常规“单次交叉拟合”估计量失效时仍能实现半参数有效性，并验证欠平滑 DCDR 估计量的渐近正态性。