Statistical inference for high-dimensional convoluted rank regression

TL;DR

This paper develops statistical inference methods for high-dimensional convoluted rank regression, addressing computational challenges and enabling confidence interval construction through novel estimators and bootstrap procedures.

Contribution

It introduces a debiased estimator and Gaussian approximation techniques for inference in high-dimensional convoluted rank regression, overcoming non-smoothness issues.

Findings

Estimation error bounds under weaker predictor conditions

Debiased estimator with Bahadur representation

Valid bootstrap procedure for confidence intervals

Abstract

High-dimensional penalized rank regression is a powerful tool for modeling high-dimensional data due to its robustness and estimation efficiency. However, the non-smoothness of the rank loss brings great challenges to the computation. To solve this critical issue, high-dimensional convoluted rank regression has been recently proposed, introducing penalized convoluted rank regression estimators. However, these developed estimators cannot be directly used to make inference. In this paper, we investigate the statistical inference problem of high-dimensional convoluted rank regression. The use of U-statistic in convoluted rank loss function presents challenges for the analysis. We begin by establishing estimation error bounds of the penalized convoluted rank regression estimators under weaker conditions on the predictors. Building on this, we further introduce a debiased estimator and provide its Bahadur representation. Subsequently, a high-dimensional Gaussian approximation for the maximum deviation of the debiased estimator is derived, which allows us to construct simultaneous confidence intervals. For implementation, a novel bootstrap procedure is proposed and its theoretical validity is also established. Finally, simulation and real data analysis are conducted to illustrate the merits of our proposed methods.