Variance component estimators OPE, NOPE and AOPE in linear mixed effects models

Optimum variance component estimation methods that yield uniformly minimum variance quadratic unbiased estimators for a full dataset are often difficult or impossible to implement. In this paper we propose an estimator which is near optimal under some distributional assumptions that can be made without specifying an exact functional form. This estimator has an exact closed form expression. An average optimum estimator, which can be used when optimum estimators exist for subsets of the data, is also proposed. Performance comparisons of the proposed estimators are made individually with other popular estimators using simulated data. A performance comparison of the average optimum estimator is made under four constraints on the variance components. A real dataset is analysed using the proposed estimators. The robustness properties of the proposed estimators, in comparison with the other estimators, including the method of moments, are also investigated, using data simulated from a skew normal distribution. The average optimum estimator is strongly robust and far superior for estimating one of the variance components, as demonstrated by making comparisons with other methods. These comparisons are based on bias, mean squared error, and mean absolute deviation. The average estimator is moderately robust in respect of the estimation of the other variance components.