1. Tracy-Widom Distribution for Heterogeneous Gram Matrices With Applications in Signal Detection.
- Author
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Ding, Xiucai and Yang, Fan
- Subjects
- *
SIGNAL detection , *RANDOM matrices , *MATRICES (Mathematics) , *ASYMPTOTIC distribution , *SYMMETRIC matrices - Abstract
Detection of the number of signals corrupted by high-dimensional noise is a fundamental problem in signal processing and statistics. This paper focuses on a general setting where the high-dimensional noise has an unknown complicated heterogeneous variance structure. We propose a sequential test which utilizes the edge singular values (i.e., the largest few singular values) of the data matrix. It also naturally leads to a consistent sequential testing estimate of the number of signals. We describe the asymptotic distribution of the test statistic in terms of the Tracy-Widom distribution. The test is shown to be accurate and have full power against the alternative, both theoretically and numerically. The theoretical analysis relies on establishing the Tracy-Widom law for a large class of Gram type random matrices with non-zero means and completely arbitrary variance profiles, which can be of independent interest. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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