No. 1064

Title:

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

Author(s):

Kuriki, Satoshi (Institute of Statistical Mathematics and Graduate University for Advanced Studies)

Key words:

Bradley-Terry model; Complex Wishart distribution; Multiple comparisons; Three-way deadlock; Tube method.

Abstract:

    Let A be a p×p real skew-symmetric Gaussian random matrix whose upper triangular element is independently distributed with the variance 1. We give the distribution of the largest singular value σ₁(A) of A in the noncentral case where the mean matrix is not necessarily 0. Moreover, by noting the fact that the largest singular value can be regarded as the maximum of a Gaussian field, we give the distribution of the standardized largest singular value σ₁(A)/(trA'A/2)^1/2 in the central case by means of the tube method. In these derivations, it is shown that the joint distribution of the singular values of A is one-to-one to that of the eigenvalues of the [p/2]×[p/2] noncentral complex Wishart distribution with p-[p/2]-1/2 degrees of freedom. These distributional results are utilized in Scheffé (1952)'s paired comparisons model. We propose tests for the hypothesis of no interaction (hypothesis of subtractivity) based on the largest singular value of the residual matrix under the null hypothesis. Professional baseball league data are analyzed as an illustrative example of paired comparisons.