Linear inference under matrix-stable errors

Abstract

Linear inference is the foundation stone for much of theoretical and applied statistics. In practice errors often have excessive tails and are lacking the moments required in conventional usage. For random vector responses such errors often are modeled via spherical α-stable distributions with stability index α ϵ (0, 2], arising in turn through central limit theory but converging to non-Gaussian limits. Earlier work [Jensen, D.R. (2018). Biom. Biostat. Int. J. 7: 205–210] reexamined conventional linear models under n-dimensional α-stable responses, to the effect that Ordinary Least Square (OLS) solutions and residual vectors under α-stable errors also have α-stable distributions, whereas F ratios remain exact in level and power as for Gaussian errors. The present study generalizes those findings to include multivariate linear models having matrix responses of order (n×k). Topics in inference focus on both location and scale matrices, the latter in connection with analogs of simple, multiple, and canonical correlations without benefit of second moments, seen nonetheless to gauge degrees of association under α-stable symmetry.