On the multivariate components of variance problem
Statistical procedures for making inferences on the variance components in univariate mixed effect models have been developed and extensively used in many fields. Development for multivariate mixed models has been relatively limited. One important issue in the multivariate problem is determining the rank of a covariance component. Testing for the rank can be considered a natural extension of the univariate problem of testing for the existence of a random effect. Knowledge on the rank can be utilized to obtain efficient estimators. Also, the true unknown rank of a covariance component influences properties of estimators of this and other covariance components. This issue on the rank is one of the underlying themes throughout this dissertation. A difficulty in developing asymptotic inference procedures for the general random effect problem is the nonexistence of a single index over which the limit is taken. For example, a one-way model has two indices, the numbers of groups and replicates. In order to develop inference procedures useful for various practical situations, asymptotic theory has to be developed under correspondingly various conditions. An eventual goal of this dissertation is to develop approximate inference procedures which can be justifiably used for a wide range of practical sampling configurations;To develop asymptotic theory, a certain nonstandard result on the limiting distribution of the roots of a determinantal equation is needed. The first paper of this dissertation presents general results on such a limiting distribution. The second paper deals with the rank testing problem. A number of asymptotic and exact procedures are discussed for a large class of multivariate mixed effect models. Practical testing procedures which can be used under various sampling configurations are derived. The third paper discusses asymptotic properties of the covariance component estimators in the multivariate one-way random effect model with possibly incorrect specification of the rank of the between-group component. Approximate inference procedures for covariance components which are useful for most practical situations are proposed.