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December 2013 Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence
Ping-Shou Zhong, Song Xi Chen, Minya Xu
Ann. Statist. 41(6): 2820-2851 (December 2013). DOI: 10.1214/13-AOS1168

Abstract

We consider two alternative tests to the Higher Criticism test of Donoho and Jin [Ann. Statist. 32 (2004) 962–994] for high-dimensional means under the sparsity of the nonzero means for sub-Gaussian distributed data with unknown column-wise dependence. The two alternative test statistics are constructed by first thresholding $L_{1}$ and $L_{2}$ statistics based on the sample means, respectively, followed by maximizing over a range of thresholding levels to make the tests adaptive to the unknown signal strength and sparsity. The two alternative tests can attain the same detection boundary of the Higher Criticism test in [Ann. Statist. 32 (2004) 962–994] which was established for uncorrelated Gaussian data. It is demonstrated that the maximal $L_{2}$-thresholding test is at least as powerful as the maximal $L_{1}$-thresholding test, and both the maximal $L_{2}$ and $L_{1}$-thresholding tests are at least as powerful as the Higher Criticism test.

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Ping-Shou Zhong. Song Xi Chen. Minya Xu. "Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence." Ann. Statist. 41 (6) 2820 - 2851, December 2013. https://doi.org/10.1214/13-AOS1168

Information

Published: December 2013
First available in Project Euclid: 17 December 2013

zbMATH: 1294.62128
MathSciNet: MR3161449
Digital Object Identifier: 10.1214/13-AOS1168

Subjects:
Primary: 62H15
Secondary: 62G20 , 62G32

Keywords: large $p$ , large deviation , optimal detection boundary , small $n$ , sparse signal , thresholding , Weak dependence

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 6 • December 2013
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