Distributions of the Largest Singular Values of Skew-Symmetric Random Matrices and their Applications to Paired Comparisons

Abstract

Let A be a real skew-symmetric Gaussian random matrix whose upper triangular elements are independently distributed according to the standard normal distribution. We provide the distribution of the largest singular value σ1 of A. Moreover, by acknowledging the fact that the largest singular value can be regarded as the maximum of a Gaussian field, we deduce the distribution of the standardized largest singular value . These distributional results are utilized in Scheffé's paired comparisons model. We propose tests for the hypothesis of subtractivity based on the largest singular value of the skew-symmetric residual matrix. Professional baseball league data are analyzed as an illustrative example.