Presentation: Doubly-robust Asymptotically Linear Z-Estimators
Speaker: David Whitney, Graduate Student, UW Biostatistics
Abstract: Doubly-robust (DR) estimators have been well-studied in statistics, particularly in the area of causal inference. These estimators remain consistent for the target of inference provided at least one of two nuisance parameters are modeled correctly. However, asymptotic linearity is not guaranteed for DR estimators under model misspecification. This is the case, for instance, when the nuisance estimators converge to their limits at slower than the parametric rate.
Previous work has shown that for substitution estimators, such as the average treatment effect, one may construct modified doubly-robust estimators which remain asymptotically linear. These previous examples all rely on scrutiny of explicitly defined remainder terms. However, many pathwise differentiable parameters are defined implicitly (e.g. as the argument minimizing some population risk). Doubly-robust M-estimators and Z-estimators of these quantities are themselves implicitly defined, precluding direct study of the remainder.
We show that it is often possible to derive doubly-robust asymptotically linear estimators even if the target parameter is defined implicitly. This result follows by study of the first order estimation remainder. As examples, we consider the counterfactual quantile process and working parametric models for adjusted treatment effects. Combining our results with the targeted minimum loss framework, we derive Z-estimators for these parameters. The performance of the proposed Z-estimators is illustrated for various types of misspecifications in a preliminary simulation study.