Goodness of fit procedures for the multivariate linear model based on spherical harmonics

The theory of the goodness of fit procedures for multivariate normality based on spherical harmonics, introduced in [6], is extended to cover the context of the multivariate linear model (MLM). The limiting distribution of the statistics considered does depend on the distribution of the covariates in the MLM, and our results provide a complete description of the manner in which the covariate distribution affects the goodness of fit statistics. We provide two methods for approximation of the limiting distributions when, as is usually the case, the covariate distribution is unknown, and evaluate their performance in simulations.