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August 2022 Optimal signal detection in some spiked random matrix models: Likelihood ratio tests and linear spectral statistics
Debapratim Banerjee, Zongming Ma
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Ann. Statist. 50(4): 1910-1932 (August 2022). DOI: 10.1214/21-AOS2150


We study signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on signal components that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-to-noise ratios are below certain bounds. In addition, the log-likelihood ratios can be asymptotically decomposed as weighted sums of a collection of statistics which we call bipartite signed cycles. Based on this decomposition, we show that below the bounds we could always achieve the asymptotically optimal powers of likelihood ratio tests via tests based on linear spectral statistics which have polynomial time complexity.

Funding Statement

The research is supported in part by NSF Career Award DMS-1352060 and a Sloan Research Fellowship.


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Debapratim Banerjee. Zongming Ma. "Optimal signal detection in some spiked random matrix models: Likelihood ratio tests and linear spectral statistics." Ann. Statist. 50 (4) 1910 - 1932, August 2022.


Received: 1 July 2018; Revised: 1 June 2021; Published: August 2022
First available in Project Euclid: 25 August 2022

Digital Object Identifier: 10.1214/21-AOS2150

Primary: 62C05 , 62F05
Secondary: 60F05

Keywords: contiguity , finite rank deformation , Principal Component Analysis , Random graphs , signal detection

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 4 • August 2022
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