Adaptive Test Procedure for High Dimensional Regression Coefficient

2026-02-08

AI Summary

This work addresses the challenge of hypothesis testing in high-dimensional regression when the underlying sparsity structure is unknown. The authors propose a unified L-statistic testing framework that adaptively aggregates the top-k strongest signals by ranking coordinate-wise evidence, thereby integrating both max-type and sum-type testing strategies. A key innovation is the Cauchy combination adaptive range test, which selects k via a dyadic grid search. The paper establishes theoretical guarantees by proving joint weak convergence and asymptotic independence between extreme-value components and normalized L-statistics. Combining extreme value theory, Cauchy combination methods, and wild bootstrap calibration, the proposed approach maintains accurate type I error control and high power under both sparse and dense alternatives, and remains valid for non-Gaussian design matrices.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Calibration & Uncertainty Quantification
• 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 evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We develop a unified $L$-statistic testing framework for high-dimensional regression coefficients that adapts to unknown sparsity. The proposed statistics rank coordinate-wise evidence measures and aggregate the top $k$ signals, bridging classical max-type and sum-type tests. We establish joint weak convergence of the extreme-value component and standardized $L$-statistics under mild conditions, yielding an asymptotic independence that justifies combining multiple $k$'s. An adaptive omnibus test is constructed via a Cauchy combination over a dyadic grid of $k$, and a wild bootstrap calibration is provided with theoretical guarantees. Simulations demonstrate accurate size and strong power across sparse and dense alternatives, including non-Gaussian designs.

Problem

• high-dimensional regression
• adaptive testing
• sparsity
• hypothesis testing
• L-statistic

Innovation

• L-statistic
• adaptive testing
• high-dimensional regression
• sparsity adaptation
• Cauchy combination