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Adaptive L-statistics for high dimensional test problem

Huifang Ma, Long Feng, Zhaojun Wang

Abstract

In this study, we focus on applying L-statistics to the high-dimensional one-sample location test problem. Intuitively, an L-statistic with $k$ parameters tends to perform optimally when the sparsity level of the alternative hypothesis matches $k$. We begin by deriving the limiting distributions for both L-statistics with fixed parameters and those with diverging parameters. To ensure robustness across varying sparsity levels of alternative hypotheses, we first establish the asymptotic independence between L-statistics with fixed and diverging parameters. Building on this, we propose a Cauchy combination test that integrates L-statistics with different parameters. Both simulation results and real-data applications highlight the advantages of our proposed methods.

Adaptive L-statistics for high dimensional test problem

Abstract

In this study, we focus on applying L-statistics to the high-dimensional one-sample location test problem. Intuitively, an L-statistic with parameters tends to perform optimally when the sparsity level of the alternative hypothesis matches . We begin by deriving the limiting distributions for both L-statistics with fixed parameters and those with diverging parameters. To ensure robustness across varying sparsity levels of alternative hypotheses, we first establish the asymptotic independence between L-statistics with fixed and diverging parameters. Building on this, we propose a Cauchy combination test that integrates L-statistics with different parameters. Both simulation results and real-data applications highlight the advantages of our proposed methods.

Paper Structure

This paper contains 16 sections, 153 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. One sample location problem
  3. The test with fixed $k$
  4. The test with diverging $k=\lceil\gamma p\rceil$
  5. Adaptive test
  6. Simulation
  7. Real data application
  8. Conclusion
  9. Appendix
  10. Proof of (i) in Theorem \ref{['asymptotic distribution of ordered t_i']}
  11. Proof of (ii) in Theorem \ref{['asymptotic distribution of ordered t_i']}
  12. Proof of Theorem \ref{['power of T_k']}
  13. Proof of Theorem \ref{['asymptotic normality']}
  14. Proof of (i) in Theorem \ref{['asymptotic independence']}
  15. Proof of (ii) in Theorem \ref{['asymptotic independence']}
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Power of individual L-tests with different numbers of nonzero means.
  • Figure 2: Power of three adaptive tests with different numbers of nonzero means.
  • Figure 3: Power of $T_1,T_5,T_p,T_{\lceil 0.5p\rceil}$ tests with different numbers of nonzero means.
  • Figure 4: Histogram of $t$-test statistics and corresponding $p$-values of each security.