The Annals of Statistics

Innovated higher criticism for detecting sparse signals in correlated noise

Peter Hall and Jiashun Jin

Full-text: Open access

Abstract

Higher criticism is a method for detecting signals that are both sparse and weak. Although first proposed in cases where the noise variables are independent, higher criticism also has reasonable performance in settings where those variables are correlated. In this paper we show that, by exploiting the nature of the correlation, performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer. Indeed, it turns out that the case of independent noise is the most difficult of all, from a statistical viewpoint, and that more accurate signal detection (for a given level of signal sparsity and strength) can be obtained when correlation is present. We characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary. The boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is Toeplitz.

Article information

Source
Ann. Statist., Volume 38, Number 3 (2010), 1686-1732.

Dates
First available in Project Euclid: 24 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1269452652

Digital Object Identifier
doi:10.1214/09-AOS764

Mathematical Reviews number (MathSciNet)
MR2662357

Zentralblatt MATH identifier
1189.62080

Subjects
Primary: 62G10: Hypothesis testing 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G32: Statistics of extreme values; tail inference 62H15: Hypothesis testing

Keywords
Adding noise Cholesky factorization empirical process innovation multiple hypothesis testing sparse normal means spectral density Toeplitz matrix

Citation

Hall, Peter; Jin, Jiashun. Innovated higher criticism for detecting sparse signals in correlated noise. Ann. Statist. 38 (2010), no. 3, 1686--1732. doi:10.1214/09-AOS764. https://projecteuclid.org/euclid.aos/1269452652


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