Rank-based Maxsum test for high dimensional regression coefficient

2026-03-15

AI Summary

This study addresses the challenge of global inference on regression coefficients in high-dimensional linear models, where existing methods often lack robustness under heavy-tailed errors or fail to adapt when the sparsity structure is unknown. The authors propose a rank-based max-sum test that integrates the Wilcoxon rank-sum statistic with the maximum-type statistic. They establish, for the first time, the joint asymptotic distribution of these two statistics under the null hypothesis and exploit their asymptotic independence to combine p-values via the Cauchy combination method. This approach requires no light-tailed error assumption and automatically adapts to both dense and sparse alternative scenarios. Extensive simulations demonstrate that the proposed test accurately controls Type I error across various error distributions and sparsity levels while achieving substantially higher statistical power.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study global inference for regression coefficients in high-dimensional linear models under potentially heavy-tailed errors. While sum-type tests are powerful for dense alternatives and max-type tests excel for sparse alternatives, practical applications rarely reveal the sparsity level, and many existing procedures rely on light-tail assumptions. Motivated by the Wilcoxon-score sum test of Feng et al. (2013) and the two Wilcoxon-score maximum tests of Xu and Zhou (2021), we establish under $H_0$ the asymptotic independence between the rank-based sum statistic and each max statistic. These joint limit results justify principled $p$-value aggregation, and we propose two adaptive rank-based maxsum tests via the Cauchy combination method (Liu and Xie, 2020). The proposed procedures inherit robustness from rank-based construction and adaptivity from combining dense- and sparse-sensitive components. Simulation studies confirm accurate size control and strong power across a wide range of error distributions and sparsity regimes.

Problem

• high-dimensional regression
• heavy-tailed errors
• global inference
• sparsity
• rank-based test

Innovation

• rank-based test
• high-dimensional inference
• maxsum test
• Cauchy combination
• heavy-tailed errors