Model-Assisted Uniformly Honest Inference for Optimal Treatment Regimes in High Dimension

TL;DR

This paper introduces a new statistical framework for high-dimensional decision-making that provides honest, uniform inference on variable relevance for optimal treatment regimes, accommodating complex interactions and nonconvex estimation challenges.

Contribution

It develops a novel approach using local restricted strong convexity and a wild bootstrap method for honest inference in high-dimensional semiparametric models with complex interactions.

Findings

The proposed method achieves near-oracle estimation error bounds.

The wild bootstrap provides asymptotically honest uniform inference.

Simulation results show satisfactory performance in high-dimensional settings.

Abstract

This paper develops new tools to quantify uncertainty in optimal decision making and to gain insight into which variables one should collect information about given the potential cost of measuring a large number of variables. We investigate simultaneous inference to determine if a group of variables is relevant for estimating an optimal decision rule in a high-dimensional semiparametric framework. The unknown link function permits flexible modeling of the interactions between the treatment and the covariates, but leads to nonconvex estimation in high dimension and imposes significant challenges for inference. We first establish that a local restricted strong convexity condition holds with high probability and that any feasible local sparse solution of the estimation problem can achieve the near-oracle estimation error bound. We further rigorously verify that a wild bootstrap procedure based on a debiased version of the local solution can provide asymptotically honest uniform inference for the effect of a group of variables on optimal decision making. The advantage of honest inference is that it does not require the initial estimator to achieve perfect model selection and does not require the zero and nonzero effects to be well-separated. We also propose an efficient algorithm for estimation. Our simulations suggest satisfactory performance. An example from a diabetes study illustrates the real application.