## The Annals of Statistics

### Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence

#### 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.

#### Article information

Source
Ann. Statist., Volume 41, Number 6 (2013), 2820-2851.

Dates
First available in Project Euclid: 17 December 2013

https://projecteuclid.org/euclid.aos/1387313391

Digital Object Identifier
doi:10.1214/13-AOS1168

Mathematical Reviews number (MathSciNet)
MR3161449

Zentralblatt MATH identifier
1294.62128

Subjects
Primary: 62H15: Hypothesis testing
Secondary: 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference

#### Citation

Zhong, Ping-Shou; Chen, Song Xi; Xu, Minya. Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence. Ann. Statist. 41 (2013), no. 6, 2820--2851. doi:10.1214/13-AOS1168. https://projecteuclid.org/euclid.aos/1387313391

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#### Supplemental materials

• Supplementary material: A supplement to “Tests alternative to higher criticism for high-dimensional means under sparsity and column-wise dependence”. The supplementary material contains proofs for Proposition 1 and Theorem 1 in Section 2.