The Annals of Statistics

Two-sample and ANOVA tests for high dimensional means

Song Xi Chen, Jun Li, and Ping-Shou Zhong

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This paper considers testing the equality of two high dimensional means. Two approaches are utilized to formulate $L_{2}$-type tests for better power performance when the two high dimensional mean vectors differ only in sparsely populated coordinates and the differences are faint. One is to conduct thresholding to remove the nonsignal bearing dimensions for variance reduction of the test statistics. The other is to transform the data via the precision matrix for signal enhancement. It is shown that the thresholding and data transformation lead to attractive detection boundaries for the tests. Furthermore, we demonstrate explicitly the effects of precision matrix estimation on the detection boundary for the test with thresholding and data transformation. Extension to multi-sample ANOVA tests is also investigated. Numerical studies are performed to confirm the theoretical findings and demonstrate the practical implementations.

Article information

Ann. Statist., Volume 47, Number 3 (2019), 1443-1474.

Received: June 2016
Revised: July 2017
First available in Project Euclid: 13 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing
Secondary: 62E20: Asymptotic distribution theory

ANOVA data transformation large $p$ small $n$ sparse signals thresholding two-sample tests for means


Chen, Song Xi; Li, Jun; Zhong, Ping-Shou. Two-sample and ANOVA tests for high dimensional means. Ann. Statist. 47 (2019), no. 3, 1443--1474. doi:10.1214/18-AOS1720.

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Supplemental materials

  • Supplement to “Two-sample and ANOVA tests for high dimensional means”. The Supplementary Material provides the proofs of lemmas, propositions and Theorems 2, 3 and 5. It also includes extra simulation results and an empirical study.