On the Optimality of Robust Inference on the Mean Outcome under Optimal Treatment Regime

Abstract

When an optimal treatment regime (OTR) is considered, we need to address the question of how good the OTR is in a valid and efficient way. The classical statistical inference applied to the mean outcome under the OTR, assuming the OTR is the same as the estimated OTR, might be biased when the regularity assumption that the OTR is unique is violated. Although several methods have been proposed to allow nonregularity in inference on the mean outcome under the OTR, the optimality of such inference is unclear due to challenges in deriving semiparametric efficiency bounds under potential nonregularity. In this paper, we address the bias issue induced by potential nonregularity via adaptive smoothing over the estimated OTR and develop a valid inference procedure on the mean outcome under the OTR regardless of whether the regularity assumption is satisfied or not. We establish the optimality of the proposed method by deriving a lower bound of the asymptotic variance for the robust asymptotically linear unbiased estimator to the mean outcome under the OTR and showing that our proposed estimator achieves the variance lower bound. The considered class of the estimator is general and includes the efficient regular estimator and the current state-of-the-art approach allowing nonregularity, and the derived lower bound of the asymptotic variance can be viewed as an extension of the classical semiparametric theory for OTR to a more general scenario allowing nonregularity. The merit of the proposed method is demonstrated by re-analyzing the ACTG 175 trial.

• Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China

Keywords: Adaptive smoothing – Bias correction – Nonregularity – Robust asymptotically linear unbiased estimators