UC Riverside

This dissertation aims at searching for the optimum variance components estimates in the mixed-effects model. Traditional estimation methods of the variance components include the analysis of variance/method of moment (ANOVA/MoM) estimation, which is the optimum estimation (OPE) when the data are balanced, the maximum likelihood estimation (MLE) and the restricted maximum likelihood estimation (REMLE). However, when the data have small sample sizes and unbalanced structures, the optimum estimates do not exist, ML estimates are biased, MLE and REMLE cannot provide the closed-form expressions of the estimates to study their small-sample statistical properties. To solve those problems, we proposed the near optimum estimation (NOPE) method and the average optimum estimation (AOPE) method when the data are unbalanced in DOE. When estimating the variance components and a linear function of variance components in SAE, we proposed methods of finding the unbiased quadratic estimators with smaller variances than the corresponding MoM estimators. We presented simulation studies to evaluate the estimation performance of our proposed methods and compare them with MoM, ML and REML. All of our proposed estimators have closed-form expressions and do not require the functional form in the distributional assumptions.