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 paper 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 data set.

Multivariate-Sign-Based High-Dimensional Tests for the Two-Sample Location Problem

“…Please refer to Appendix C in the Supplemental Material for the closed-form of this bias and associated asymptotic analysis. In fact, the test statistic proposed by Wang et al (2015) is essentially in a similar fashion to Q 2 n . However, their method does not suffer from additional biases because in a one-sample problem we do not need the estimate of spatial median.…”

“…Simulation comparisons show that our procedure performs reasonably well in terms of sizes and power for a wide range of dimensions, sample sizes and distributions. Recently, Wang et al (2015) proposed a high-dimensional nonparametric multivariate test for mean vector based on spatial-signs.…”

A simpler spatial-sign-based two-sample test for high-dimensional data

“…Recently, Wang et al [13] and Feng et al [4] proposed high-dimensional spatial-sign-based tests for the one-sample and two-sample problems, respectively. The test statistic proposed in the former one is essentially in a similar fashion to Chen and Qin's [3] test statistic, it is but not directly applicable for the two-sample problem due to the bias from estimating the location parameter.…”

Variable Selection via Partial Correlation

“…The elliptical distribution family contains a much broader class of distributions than the normal distribution family, such as mixture of normal distributions, multivariate t-distribution, multi-uniform distribution on unit sphere, Pearson Type II distribution, among others. It has been used as a tool to study the robustness of normality in the literature of multivariate nonparametric tests (Mottonen, Oja and Tienari, 1997; Oja and Randles, 2004; Chen, Wiesel and Hero, 2011; Soloveychik and Wiesel, 2015; Wang, Peng and Li, 2015). The elliptical linear regressions have been proposed in Osiewalski (1991); Osiewalski and Steel (1993); Arellano-Valle, del Pino and Iglesias (2006); Fan and Lv (2008); Liang and Li (2009); Vidal and Arellano-Valle (2010), and have received more and more attentions in the recent literature (Arellano-Valle, del Pino and Iglesias, 2006; Fan and Lv, 2008; Liang and Li, 2009; Vidal and Arellano-Valle, 2010).…”