Linear inference under alpha–stable errors

Abstract

Linear inference remains pivotal in statistical practice, despite errors often having excessive tails and thus deficient of moments required in conventional usage. Such errors are modeled here via spherical α-stable measures on n  with stability index (0,2], α∈ arising in turn through multivariate central limit theory devoid of the second moments required for Gaussian limits. This study revisits linear inference under α-stable errors, focusing on aspects to be salvaged from the classical theory even without moments. Critical entities include Ordinary Least Squares () OLS solutions, residuals, and conventional F ratios in inference. Closure properties are seen in that 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. Although correlations are undefined for want of second moments, corresponding scale parameters are seen to gauge degrees of association under α-stable symmetry.