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When to initiate treatment on patients is an important problem in manymedical studies such as AIDS and cancer. In this article, we formulate thetreatment initiation time problem for time-to-event data and propose an optimalindividualized regime that determines the best treatment initiation time forindividual patients based on their characteristics. Different from existingoptimal treatment regimes where treatments are undertaken at a pre-specifiedtime, here new challenges arise from the complicated missing mechanisms intreatment initiation time data and the continuous treatment rule in terms ofinitiation time. To tackle these challenges, we propose to use restricted meanresidual lifetime as a value function to evaluate the performance of differenttreatment initiation regimes, and develop a nonparametric estimator for thevalue function, which is consistent even when treatment initiation times arenot completely observable and their distribution is unknown. We also establishthe asymptotic properties of the resulting estimator in the decision rule andits associated value function estimator. In particular, the asymptoticdistribution of the estimated value function is nonstandard, which follows aweighted chi-squared distribution. The finite-sample performance of theproposed method is evaluated by simulation studies and is further illustratedwith an application to a breast cancer data.

On Estimating Optimal Regime for Treatment Initiation Time Based on Restricted Mean Residual Lifetime