## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 2 (2017), 771-799.

### Tests for high-dimensional data based on means, spatial signs and spatial ranks

Anirvan Chakraborty and Probal Chaudhuri

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 2 (2017), 771-799.

**Dates**

Received: May 2015

Revised: March 2016

First available in Project Euclid: 16 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1494921957

**Digital Object Identifier**

doi:10.1214/16-AOS1467

**Mathematical Reviews number (MathSciNet)**

MR3650400

**Zentralblatt MATH identifier**

1368.62147

**Subjects**

Primary: 62H15: Hypothesis testing 62G10: Hypothesis testing

Secondary: 60G10: Stationary processes 62E20: Asymptotic distribution theory

**Keywords**

ARMA processes heavy tailed distributions permutation tests $\rho$-mixing randomly scaled $\rho$-mixing spherical distributions stationary sequences

#### Citation

Chakraborty, Anirvan; Chaudhuri, Probal. Tests for high-dimensional data based on means, spatial signs and spatial ranks. Ann. Statist. 45 (2017), no. 2, 771--799. doi:10.1214/16-AOS1467. https://projecteuclid.org/euclid.aos/1494921957

#### Supplemental materials

- Supplement to “Tests for high-dimensional data based on means, spatial signs and spatial ranks”. This supplemental article contains additional mathematical details related to the proof of part (a) of Theorem 3.3 and the detailed results of the simulation study done in Section 5 of the paper.Digital Object Identifier: doi:10.1214/16-AOS1467SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.