Distributions of the largest singular values of skew-symmetric random matrices and their applications to paired comparisons

TL;DR

This paper derives the distribution of the largest singular value of skew-symmetric Gaussian matrices and applies these results to hypothesis testing in paired comparisons, demonstrated with baseball data.

Contribution

It provides the first explicit distributional results for the largest singular value of skew-symmetric Gaussian matrices and applies them to statistical tests in paired comparisons.

Findings

Distribution of the largest singular value derived

Distribution of standardized largest singular value obtained

Applied to hypothesis testing in paired comparisons with real data

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.