  Personalized decision-making, tailored to individual characteristics, is
gaining significant attention. The optimal treatment regime aims to provide the
best-expected outcome in the entire population, known as the value function.
One approach to determine this optimal regime is by maximizing the Augmented
Inverse Probability Weighting (AIPW) estimator of the value function. However,
the derived treatment regime can be intricate and nonlinear, limiting their
use. For clarity and interoperability, we emphasize linear regimes and
determine the optimal linear regime by optimizing the AIPW estimator within set
constraints.
  While the AIPW estimator offers a viable path to estimating the optimal
regime, current methodologies predominantly focus on its asymptotic
distribution, leaving a gap in studying the linear regime itself. However,
there are many benefits to understanding the regime, as pinpointing significant
covariates can enhance treatment effects and provide future clinical guidance.
In this paper, we explore the asymptotic distribution of the estimated linear
regime. Our results show that the parameter associated with the linear regime
follows a cube-root convergence to a non-normal limiting distribution
characterized by the maximizer of a centered Gaussian process with a quadratic
drift. When making inferences for the estimated linear regimes with cube-root
convergence in practical scenarios, the standard nonparametric bootstrap is
invalid. As a solution, we facilitate the Cattaneo et al. (2020) bootstrap
technique to provide a consistent distributional approximation for the
estimated linear regimes, validated further through simulations and real-world
data applications from the eICU Collaborative Research Database.
