The Distribution of the Covariance Matrix for a Subset of Elliptical Distributions with Extension to Two Kurtosis Parameters

We address the important practical problem of selecting covariates in mixed linear models when the covariance structure is known from the data collection process and there are a possibly large number of covariates available. In particular, we consider procedures which can be considered extensions of the analysis of deviance to mixed linear models. This approach provides an alternative to likelihood ratio test methodology which can be applied in the case that the components of variance are estimated by restricted maximum likelihood (REML), thus resolving the open question of how to proceed in this context. Moreover, it is simple to robustify and allows us to consider a wider class of procedures than those which fit into the simple likelihood ratio test framework. The key insights are that the deviance should be specified by the procedure used to estimate the fixed effects and that the estimated covariance matrix should be held fixed across different models for the fixed effects.