Nowadays, more literature estimates their parameters of interest relying on estimating equations with two or more nuisance parameters. In some cases, one might be able to find a population-level doubly (or possibly multiply) robust estimating equation which has zero mean provided one of the nuisance parameters is correctly specified, without knowing which. This property is appealing in practice because it suggests "model doubly robust" estimators that entail extra protection against model misspecification. Typically asymptotic inference of such a doubly robust estimator is relatively simple through classical Z-estimation theory under standard regularity conditions. In other cases, machine learning techniques are leveraged to achieve "rate double robustness", with cross fitting. However, the classical theory might be insufficient when all nuisance parameters involve complex time structures and are possibly in the form of continuous-time stochastic nuisance processes. In such cases, we caution that extra assumptions are needed, especially on total variation. In this paper, as an example, we consider a general class of double robust estimating equations and develop generic assumptions on the asymptotic properties of the estimators of nuisance parameters such that the resulted estimator for the parameter of interest is consistent and asymptotically normal. We illustrate our framework in some examples. We also caution a gap between population double robustness and rate double robustness.

翻译：如今，越来越多的文献依赖于包含两个或更多 nuisance 参数的估计方程来估计其感兴趣的参数。在某些情况下，人们可能能够找到一种在总体层面具有双（或可能多）稳健性的估计方程，只要其中一个 nuisance 参数被正确指定（无需知道是哪一个），该方程的均值就为零。这一性质在实践中颇具吸引力，因为它提供了“模型双稳健”估计量，从而对模型误设提供额外保护。通常，在标准正则性条件下，通过经典的 Z-估计理论，这种双稳健估计量的渐近推断相对简单。在其他情况下，则利用机器学习技术并结合交叉拟合来实现“率双稳健性”。然而，当所有 nuisance 参数涉及复杂的时间结构，并且可能以连续时间随机 nuisance 过程的形式存在时，经典理论可能并不充分。在这种情况下，我们提醒需要附加假设，特别是关于总变差的假设。本文以一类一般的双稳健估计方程为例，针对 nuisance 参数估计量的渐近性质提出了一类通用假设，使得由此得到的感兴趣参数估计量具有一致性和渐近正态性。我们通过一些例子来说明我们的框架。同时，我们也提醒注意总体双稳健性与率双稳健性之间的差异。