The bootstrap is a popular method of constructing confidence intervals due to its ease of use and broad applicability. Theoretical properties of bootstrap procedures have been established in a variety of settings. However, there is limited theoretical research on the use of the bootstrap in the context of estimation of a differentiable functional in a nonparametric or semiparametric model when nuisance functions are estimated using machine learning. In this article, we provide general conditions for consistency of the bootstrap in such scenarios. Our results cover a range of estimator constructions, nuisance estimation methods, bootstrap sampling distributions, and bootstrap confidence interval types. We provide refined results for the empirical bootstrap and smoothed bootstraps, and for one-step estimators, plug-in estimators, empirical mean plug-in estimators, and estimating equations-based estimators. We illustrate the use of our general results by demonstrating the asymptotic validity of bootstrap confidence intervals for the average density value and G-computed conditional mean parameters, and compare their performance in finite samples using numerical studies. Throughout, we emphasize whether and how the bootstrap can produce asymptotically valid confidence intervals when standard methods fail to do so.

翻译：自助法因其易用性和广泛适用性而成为构建置信区间的常用方法。自助法程序的理论性质已在多种场景中得到确立。然而，当干扰函数通过机器学习估计时，在非参数或半参数模型中估计可微泛函的背景下，关于自助法使用的理论研究仍十分有限。本文为在这种场景下自助法的一致性提供了通用条件。我们的结果涵盖了多种估计量构造、干扰估计方法、自助法抽样分布以及自助法置信区间类型。我们针对经验自助法和平滑自助法，以及一步估计量、插件估计量、经验均值插件估计量和基于估计方程的估计量提供了精细化结果。通过展示自助法置信区间对于平均密度值和G计算的条件均值参数的渐近有效性，我们阐明了通用结果的应用，并通过数值研究比较了它们在小样本中的表现。自始至终，我们重点探讨当标准方法无法构建渐近有效置信区间时，自助法是否以及如何能够做到这一点。