Doubly robust estimators have gained widespread popularity in various fields due to their ability to provide unbiased estimates under model misspecification. However, the asymptotic theory for doubly robust estimators with continuous-time nuisance parameters remains largely unexplored. In this short communication, we address this gap by developing a general asymptotic theory for a class of doubly robust estimating equations involving stochastic processes and Riemann-Stieltjes integrals. We introduce generic assumptions on the nuisance parameter estimators that ensure the consistency and asymptotic normality of the resulting doubly robust estimator. Our results cover both the model doubly robust estimator, which relies on parametric or semiparametric models, and the rate doubly robust estimator, which allows for flexible machine learning methods. We discuss the implications of our findings and highlight the key differences between the continuous-time setting and the classical theory for doubly robust estimators. Our work provides a solid theoretical foundation for the use of doubly robust estimators in complex settings with continuous-time nuisance parameters, paving the way for future research and applications.

翻译：双稳健估计量因其在模型误设下仍能提供无偏估计的能力，在各领域得到了广泛应用。然而，针对连续时间冗余参数的双稳健估计量，其渐近理论仍鲜有探索。在这篇简短通讯中，我们通过发展一类涉及随机过程和黎曼-斯蒂尔杰斯积分的双稳健估计方程的通用渐近理论，填补了这一空白。我们引入了关于冗余参数估计量的通用假设，以确保所得双稳健估计量的相合性与渐近正态性。我们的结果涵盖了依赖参数或半参数模型的模型双稳健估计量，以及允许灵活使用机器学习方法的速度双稳健估计量。我们讨论了研究结果的启示，并强调了连续时间框架与经典双稳健估计理论之间的关键差异。本研究为在包含连续时间冗余参数的复杂场景中应用双稳健估计量提供了坚实的理论基础，为未来研究与应用铺平了道路。