Relaxed doubly robust estimation in causal inference

Causal inference plays a crucial role in biomedical studies and social sciences. Over the years, researchers have devised various methods to facilitate causal inference, particularly in observational studies. Among these methods, the doubly robust estimator distinguishes itself through a remarkable feature: it retains its consistency even when only one of the two components – either the propensity score model or the outcome mean model – is correctly specified, rather than demanding correctness in both simultaneously. In this paper, we focus on scenarios where semiparametric models are employed for both the propensity score and the outcome mean. Semiparametric models offer a valuable blend of interpretability akin to parametric models and the adaptability characteristic of nonparametric models. In this context, achieving correct model specification involves both accurately specifying the unknown function and consistently estimating the unknown parameter. We introduce a novel concept: the relaxed doubly robust estimator. It operates in a manner reminiscent of the traditional doubly robust estimator but with a reduced requirement for double robustness. In essence, it only mandates the consistent estimate of the unknown parameter, without requiring the correct specification of the unknown function. This means that it only necessitates a partially correct model specification. We conduct a thorough analysis to establish the double robustness and semiparametric efficiency of our proposed estimator. Furthermore, we bolster our findings with comprehensive simulation studies to illustrate the practical implications of our approach.