Statistics and Its Interface

Volume 12 (2019)

Number 1

An adaptive spatial-sign-based test for mean vectors of elliptically distributed high-dimensional data

Pages: 93 – 106

DOI: https://dx.doi.org/10.4310/SII.2019.v12.n1.a9

Authors

Bu Zhou (School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, Zhejiang, China)

Jia Guo (College of Economics and Management, Zhejiang University of Technology, Hangzhou, Zhejiang, China)

Jianwei Chen (Department of Mathematics and Statistics, San Diego State University, San Diego, California, U.S.A.; and Center of Modern Applied Statistics and Big Data, School of Statistics, Huaqiao University, Quanzhou, Fujian, China)

Jin-Ting Zhang (Department of Statistics and Applied Probability, National University of Singapore)

Abstract

Recently, a nonparametric test for mean vectors of elliptically distributed high-dimensional data has been proposed in the literature. The asymptotic normality of the test statistic under some strong assumptions is established. In practice, however, these strong assumptions may not be satisfied or hardly be checked so that the above test may not perform well in terms of size control. In this paper, we propose an adaptive spatial-sign-based test for mean vectors of elliptically distributed high-dimensional data without imposing strong assumptions. The null distribution of the proposed test statistic is shown to be a chi-squared mixture which is generally skewed. We propose to approximate the null distribution using the well-known Welch–Satterthwaite $\chi^2$-approximation. The resulting approximate distribution is able to adapt to the shape of the underlying null distribution of the proposed test statistic. Simulation studies and three real data examples demonstrate that the proposed test has a better size control than the existing nonparametric test while both tests enjoy about the same powers.

Keywords

high-dimensional data, spatial-sign-based test, Welch–Satterthwaite $\chi^2$-approximation

2010 Mathematics Subject Classification

Primary 62H15. Secondary 62G10.

Full Text (PDF format)

The work of Zhou was financially supported by First Class Discipline of Zhejiang – A (Zhejiang Gongshang University – Statistics). The work of Guo was financially supported by the Australian Research Council (ARC) Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS). The work of Zhang was financially supported by the National University of Singapore Academic Research Grant R-155-000-187-114. The authors thank the Editor, an AE and a reviewer for their insightful comments and suggestions which help improve the presentation of the paper substantially.

Received 5 March 2018

Published 26 October 2018