A High-Dimensional Nonparametric Multivariate Test for Mean Vector

Abstract

This work is concerned with testing the population mean vector of nonnormal high-dimensional multivariate data. Several tests for high-dimensional mean vector, based on modifying the classical Hotelling T² test, have been proposed in the literature. Despite their usefulness, they tend to have unsatisfactory power performance for heavy-tailed multivariate data, which frequently arise in genomics and quantitative finance. This article proposes a novel high-dimensional nonparametric test for the population mean vector for a general class of multivariate distributions. With the aid of new tools in modern probability theory, we proved that the limiting null distribution of the proposed test is normal under mild conditions when p is substantially larger than n. We further study the local power of the proposed test and compare its relative efficiency with a modified Hotelling T² test for high-dimensional data. An interesting finding is that the newly proposed test can have even more substantial power gain with large p than the traditional nonparametric multivariate test does with finite fixed p. We study the finite sample performance of the proposed test via Monte Carlo simulations. We further illustrate its application by an empirical analysis of a genomics dataset. Supplementary materials for this article are available online.

• Asymptotic relative efficiency
• High-dimensional multivariate data
• Hotelling T² test
• Nonparametric multivariate test

Additional information

Notes on contributors

Lan Wang

Lan Wang (E-mail: [email protected]) is Associate Professor, and Bo Peng is graduate student (E-mail: [email protected]), School of Statistics, University of Minnesota, Minneapolis, MN 55455. Runze Li is the corresponding author and Distinguished Professor, Department of Statistics and the Methodology Center, the Pennsylvania State University, University Park, PA 16802-2111 (E-mail: [email protected]). Wang and Peng’s research is supported by an NSF grant DMS1308960. Li’s research is supported by NIDA, NIH grants P50 DA10075 and P50 DA036107. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDA or the NIH. The authors thank the editor, the AE, and three referees for their constructive comments that help us significantly improve the article. The authors also thank Professor Tiefeng Jiang for helpful discussions.

Bo Peng

Lan Wang (E-mail: [email protected]) is Associate Professor, and Bo Peng is graduate student (E-mail: [email protected]), School of Statistics, University of Minnesota, Minneapolis, MN 55455. Runze Li is the corresponding author and Distinguished Professor, Department of Statistics and the Methodology Center, the Pennsylvania State University, University Park, PA 16802-2111 (E-mail: [email protected]). Wang and Peng’s research is supported by an NSF grant DMS1308960. Li’s research is supported by NIDA, NIH grants P50 DA10075 and P50 DA036107. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDA or the NIH. The authors thank the editor, the AE, and three referees for their constructive comments that help us significantly improve the article. The authors also thank Professor Tiefeng Jiang for helpful discussions.

Runze Li

Lan Wang (E-mail: [email protected]) is Associate Professor, and Bo Peng is graduate student (E-mail: [email protected]), School of Statistics, University of Minnesota, Minneapolis, MN 55455. Runze Li is the corresponding author and Distinguished Professor, Department of Statistics and the Methodology Center, the Pennsylvania State University, University Park, PA 16802-2111 (E-mail: [email protected]). Wang and Peng’s research is supported by an NSF grant DMS1308960. Li’s research is supported by NIDA, NIH grants P50 DA10075 and P50 DA036107. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDA or the NIH. The authors thank the editor, the AE, and three referees for their constructive comments that help us significantly improve the article. The authors also thank Professor Tiefeng Jiang for helpful discussions.