Improved Estimators of Variance Components with Smaller Probability of Negativity

Abstract

SUMMARY A linear model with two variance components is considered, one variance component (say, σ 1 2 ≥ 0) corresponding to a random effect, and a second variance component (say, σ 2 > 0) corresponding to the experimental errors. A class of invariant quadratic estimators (IQEs) is characterized, having uniformly smaller mean-squared error (MSE), and uniformly smaller probability of negativity, compared with the analysis-of-variance (ANOVA) estimator of σ 1 2. It turns out that for balanced models, among IQEs of σ 1 2 with uniformly smaller MSE than its ANOVA estimator, there exists an IQE with the smallest probability of being negative, uniformly in the parameters. The results are applied to the balanced one-way classification model. Numerical computations show that the MSE improvement achievable through the use of the proposed estimators is quite significant. It is noted that, for the nonnegative estimation of σ 1 2, truncation of one of the proposed estimators at 0 provides a satisfactory solution. Extension of the results to the unbalanced case, and to the general mixed model with balanced data, is also indicated.