Inference for Variance–Covariance Parameters

In Chaps. 11 and 13 , we obtained BLUEs for estimable linear functions under the Aitken model and BLUPs for predictable linear functions under the prediction-extended Aitken model, and we noted that this methodology could be used to estimate estimable linear functions or predict predictable linear functions of β, b, and d in the mixed (and random) effects model, or more generally to estimate estimable and predict predictable linear functions in a general mixed linear model, provided that the variance–covariance parameters ψ (in the case of a mixed effects model) or θ (in the case of a general mixed model) are known. Moreover, it was also noted at the ends of both chapters that the customary procedure for performing these inferences when the variance–covariance parameters are unknown is to first estimate those parameters from the data and then use BLUE/BLUP formulas with the estimates substituted for the unknown true values. It is natural, then, to ask how the variance–covariance parameters should be estimated. Answering this question is the topic of this chapter. We begin with an answer that applies when the model is a components-of-variance model, for which a method known as quadratic unbiased estimation can be used to estimate the variance–covariance parameters, which are variance components in that case. Then we give an answer, based on likelihood-based estimation, that applies to any general mixed linear model.