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April 2017 Tests for high-dimensional data based on means, spatial signs and spatial ranks
Anirvan Chakraborty, Probal Chaudhuri
Ann. Statist. 45(2): 771-799 (April 2017). DOI: 10.1214/16-AOS1467

Abstract

Tests based on mean vectors and spatial signs and ranks for a zero mean in one-sample problems and for the equality of means in two-sample problems have been studied in the recent literature for high-dimensional data with the dimension larger than the sample size. For the above testing problems, we show that under suitable sequences of alternatives, the powers of the mean-based tests and the tests based on spatial signs and ranks tend to be same as the data dimension tends to infinity for any sample size when the coordinate variables satisfy appropriate mixing conditions. Further, their limiting powers do not depend on the heaviness of the tails of the distributions. This is in striking contrast to the asymptotic results obtained in the classical multivariate setting. On the other hand, we show that in the presence of stronger dependence among the coordinate variables, the spatial-sign- and rank-based tests for high-dimensional data can be asymptotically more powerful than the mean-based tests if, in addition to the data dimension, the sample size also tends to infinity. The sizes of some mean-based tests for high-dimensional data studied in the recent literature are observed to be significantly different from their nominal levels. This is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics. However, our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets.

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Anirvan Chakraborty. Probal Chaudhuri. "Tests for high-dimensional data based on means, spatial signs and spatial ranks." Ann. Statist. 45 (2) 771 - 799, April 2017. https://doi.org/10.1214/16-AOS1467

Information

Received: 1 May 2015; Revised: 1 March 2016; Published: April 2017
First available in Project Euclid: 16 May 2017

zbMATH: 1368.62147
MathSciNet: MR3650400
Digital Object Identifier: 10.1214/16-AOS1467

Subjects:
Primary: 62G10 , 62H15
Secondary: 60G10 , 62E20

Keywords: $\rho$-mixing , ARMA processes , heavy tailed distributions , permutation tests , randomly scaled $\rho$-mixing , spherical distributions , Stationary sequences

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 2 • April 2017
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