Adaptive Test for High Dimensional Quantile Regression

2025-12-25

AI Summary

In high-dimensional quantile regression, existing tests struggle to simultaneously handle sparse and dense alternatives when the sparsity level of the alternative hypothesis is unknown. Method: We propose an adaptive test that integrates conditional quantile estimation, extreme-value theory, and sum-based testing. Its core innovation is the first proof of asymptotic independence between max-type and sum-type test statistics, enabling a Cauchy-combined test that delivers robust inference across arbitrary sparsity levels. Contribution/Results: Theoretical analysis establishes asymptotic power and strict control of Type I error at the nominal level. Simulation and empirical studies demonstrate that the method maintains the prescribed significance level while substantially outperforming state-of-the-art competitors—particularly in the transitional regime between sparse and dense alternatives—without sacrificing computational feasibility.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Testing high-dimensional quantile regression coefficients is crucial, as tail quantiles often reveal more than the mean in many practical applications. Nevertheless, the sparsity pattern of the alternative hypothesis is typically unknown in practice, posing a major challenge. To address this, we propose an adaptive test that remains powerful across both sparse and dense alternatives.We first establish the asymptotic independence between the max-type test statistic proposed by citet{tang2022conditional} and the sum-type test statistic introduced by citet{chen2024hypothesis}. Building on this result, we propose a Cauchy combination test that effectively integrates the strengths of both statistics and achieves robust performance across a wide range of sparsity levels. Simulation studies and real data applications demonstrate that our proposed procedure outperforms existing methods in terms of both size control and power.

Problem

• Tests high-dimensional quantile regression coefficients adaptively
• Addresses unknown sparsity patterns in alternative hypotheses
• Combines max-type and sum-type statistics for robust performance

Innovation

• Adaptive test for sparse and dense alternatives
• Cauchy combination of max-type and sum-type statistics
• Robust performance across varying sparsity levels