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 $\sigma_1$ 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 $\sigma_1/\sqrt{\mathrm{tr}(A'A)/2}$. These distributional results are utilized in Scheff\'{e}'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.