A Cautionary Note on Doubly Robust Estimators Involving Continuous-time Structure

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.