Minimax Rate-Optimal Inference for Individualized Quantile Treatment Effects in High-dimensional Models
Jiachen Sun, Yin Xia
TL;DR
The paper addresses the challenge of inferring individualized quantile treatment effects (IQTE) in high-dimensional settings, capturing both personalized heterogeneity and quantile-level heterogeneity. It develops a debiased estimator for the IQTE that incorporates a variance-enhancement projection, enabling uniform Gaussian-process convergence across quantile levels and valid inference without structural assumptions on the loading vector. The authors establish asymptotic normality, construct confidence intervals, and devise hypothesis tests with minimax optimality for both interval length and detection boundaries, supported by simulations and NHANES data analyses. The work provides a principled framework for reliable, individualized treatment effect assessment in complex applications with many covariates and heterogeneous responses, with potential extensions to convolution-type methods and federated settings.
Abstract
The quantification of treatment effects plays an important role in a wide range of applications, including policy making and bio-pharmaceutical research. In this article, we study the quantile treatment effect (QTE) while addressing two specific types of heterogeneities: (a) personalized heterogeneity, which captures the varying treatment effects for different individuals, and (b) quantile heterogeneity, which accounts for how the impact of covariates varies across different quantile levels. A well-designed debiased estimator for the individualized quantile treatment effect (IQTE) is proposed to capture such heterogeneities effectively. We show that this estimator converges weakly to a Gaussian process as a function of the quantile levels and propose valid statistical inference methods, including the construction of confidence intervals and the development of hypothesis testing decision rules. In addition, the minimax optimality frameworks for these inference procedures are established. Specifically, we derive the minimax optimal rates for the expected length of confidence intervals and the magnitude of the detection boundary for hypothesis testing procedures, illustrating the superiority of the proposed estimator. The effectiveness of our methods is demonstrated through extensive simulations and an analysis of the National Health and Nutrition Examination Survey (NHANES) datasets.