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

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 ${\sigma }_1\mathrm{/}\sqrt{\mathrm{t}\mathrm{r}(A^{\mathrm{\prime }}A)\mathrm{/}2}$. 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.