Statistical Inference for Optimal Treatment Regime and Related Problems

Abstract

Precision medicine is an innovative practice for disease treatment that takes into account individual variability in genes, environment, and lifestyle for each patient. Its main aim is to estimate and make inference about the optimal treatment regime. Though many successful estimation strategies have been developed, studies on statistical inference have not attracted much attention until recently. In this thesis, we attempt to study several statistical inference problems about the optimal treatment regime and some related problems in precision medicine. My thesis is composed of three parts. In the first part, we follow a nonparametric setup to estimate the optimal treatment regime, and propose a resampling approach for inference. The estimator based on a smoothed value function significantly saves the computational cost, provides adorable theoretical properties, and ensures the validity of resampling procedures. In the second part, we adopt a semiparametric model-assisted approach, and investigate inference about the effect of a group of variables on the optimal decision rule in the high-dimensional setting. Its theoretical properties are rigorously justified, and the proposed algorithm ensures its computational efficiency. The last part introduces a new approach for estimating a high-dimensional error-in-variable regression model. It enjoys the same computational convenience of standard Dantzig estimator in the non-contamination case and requires no additional tuning parameter. Theoretically, we derive its estimation error bound. The computational efficiency of all the proposed estimation and inference procedures are demonstrated by numerical studies.