When an optimal treatment regime (OTR) is considered, we need to evaluate the OTR in a valid and efficient way. The classical inference applied to the mean outcome under OTR, assuming the OTR is the same as the estimated OTR, might be biased when the regularity assumption that OTR is unique is violated. Although several methods have been proposed to allow nonregularity in such inference, its optimality is unclear due to challenges in deriving semiparametric efficiency bounds under potential nonregularity. In this paper, we address the bias issue via adaptive smoothing over the estimated OTR and develop a valid inference procedure on the mean outcome under OTR regardless of whether regularity is satisfied. 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 OTR and showing that our proposed estimator achieves the variance lower bound. The considered estimator class is general and the derived variance lower bound paves a novel way to establish efficiency optimality theories for OTR in a more general scenario allowing nonregularity. The merit of the proposed method is demonstrated by re-analyzing the ACTG 175 trial.

翻译：当考虑最优治疗策略（OTR）时，需要以有效且合理的方式对其进行评估。传统推断方法假设OTR与估计的OTR相同，并在此基础上对OTR下的平均结局进行推断，但当OTR唯一性的正则性假设被违反时，该方法可能存在偏倚。尽管已有多种方法被提出以允许此类推断中的非正则性，但由于在潜在非正则性下推导半参数效率界存在困难，其最优性尚不明确。本文通过自适应平滑估计的OTR来解决偏倚问题，并开发了一种无论正则性条件是否满足均能对OTR下平均结局进行有效推断的方法。通过推导稳健渐近线性无偏估计量在OTR下平均结局的渐近方差下界，并证明所提估计量达到该方差下界，我们建立了所提方法的最优性。所考虑的估计量类别具有一般性，且推导的方差下界为在允许非正则性的更一般场景下建立OTR的效率最优性理论开辟了新途径。通过重新分析ACTG 175试验，我们验证了所提方法的优越性。