Adaptive Test Procedure for High Dimensional Regression Coefficient

TL;DR

This paper introduces an adaptive $L$-statistic testing framework for high-dimensional regression coefficients that effectively handles unknown sparsity levels, combining classical max and sum tests with bootstrap calibration.

Contribution

It proposes a unified, adaptive testing procedure that combines multiple $k$-based statistics using a Cauchy combination, with theoretical guarantees and practical effectiveness.

Findings

Accurate size control in simulations.

Strong power against sparse and dense alternatives.

Effective in non-Gaussian design settings.

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.