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Rank-based Maxsum test for high dimensional regression coefficient

Ping Zhao, Liangliang Yuan

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.

Rank-based Maxsum test for high dimensional regression coefficient

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 the asymptotic independence between the rank-based sum statistic and each max statistic. These joint limit results justify principled -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.
Paper Structure (7 sections, 2 theorems, 53 equations, 2 figures, 1 table)

This paper contains 7 sections, 2 theorems, 53 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Rank-based Maxsum Tests
  3. Simulation
  4. Conclusion
  5. Appendix
  6. Proof of Theorem \ref{['th1']}
  7. Proof of Theorem \ref{['th2']}

Key Result

Theorem 1

Assume conditions (C1)-(C3) hold. Under $H_0$, if $\log p=o(n^{1/5})$, we have where $\Phi(x)$ is the c.d.f of $N(0,1)$ and $F(y)=\exp(-\frac{1}{\sqrt{\pi}}e^{-y/2})$ is the c.d.f of Gumble distribution.

Figures (2)

  • Figure 1: Empirical power as a function of the number of nonzero coefficients of $\bm \beta$ under different error distributions.
  • Figure 2: Empirical power as a function of the number of nonzero coefficients of $\bm \theta$ under different error distributions.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2