Roy’s largest root under rank-one perturbations: The complex valued case and applications

• Published in: Journal of multivariate analysis Vol. 174; p. 104524
• Main Authors: Dharmawansa, Prathapasinghe, Nadler, Boaz, Shwartz, Ofer
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 01.11.2019
• Subjects: Complex Wishart distribution; Rank-one perturbation; Roy’s largest root; Signal detection in noise; secondary; Complex Wishart distribution; Rank-one perturbation; Signal detection in noise; primary; Roy’s largest root; 33C15; Signal detection in noise 2010 MSC: Primary 60B20; Seconday 62H10
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2019.05.009

The largest eigenvalue of a single or a double Wishart matrix, both known as Roy’s largest root, plays an important role in a variety of applications. Recently, via a small noise perturbation approach with fixed dimension and degrees of freedom, Johnstone and Nadler derived simple yet accurate approximations to its distribution in the real valued case, under a rank-one alternative. In this paper, we extend their results to the complex valued case for five common single matrix and double matrix settings. In addition, we study the finite sample distribution of the leading eigenvector. We present the utility of our results in several signal detection and communication applications, and illustrate their accuracy via simulations.