Near Uniformly Minimum Variance Quadratic Unbiased Estimation of Variance Components in Mixed Effects Models
Several methods are available in literature for estimating the variance components in mixed effects models. In this thesis we consider the general mixed effects model without making any distributional assumptions. The quadratic unbiased estimators are considered for estimating the variance components. The uniformly minimum variance quadratic unbiased estimation (UMVQUE) of variance components is investigated for the data obtained from both balanced and unbalanced designs. In spite of its attractive properties, the UMVQUE may not always be possible. When the UMVQUE is not possible, we propose two alternative methods for estimating the variance components. We first introduce a method of near uniformly minimum variance quadratic unbiased estimation (NUMVQUE) for an unbalanced incomplete block design. When the UMVQUE of variance components is not possible for a design with replicated blocks but it is possible with a single replication of blocks, we propose another method of average uniformly minimum variance quadratic unbiased estimation (AUMVQUE). The maximum likelihood estimation (MLE) and restricted maximum likelihood estimation (REMLE) are likelihood based procedures and therefore require the distributional assumptions to estimate the variance components. We present a simulation study to evaluate the performance of our proposed estimation methods and compare them with MLE and REMLE.