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Roy’s largest root under rank-one perturbations: the complex valued case and applications. (English) Zbl 1425.60011

Summary: 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.

MSC:

60B20 Random matrices (probabilistic aspects)
62H10 Multivariate distribution of statistics
33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\)
15B52 Random matrices (algebraic aspects)
62E20 Asymptotic distribution theory in statistics
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