A Comparison of Variance Component Estimates for Arbitrary Underlying Distributions

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Abstract
Large-sample covariance matrices for the analysis of variance (ANOVA), minimum norm quadratic unbiased estimator (MINQUE), restricted maximum likelihood (REML), and maximum likelihood (ML) estimates of variance components are presented for the unbalanced one-way model when the underlying distributions are not necessarily normal. The limiting variances depend on the design sequence, on the actual values of the variance components, and on the kurtosis parameters of the underlying distributions. (The skewness parameters and other moments do not affect the limiting distributions.) Because all estimates are consistent and asymptotically normal, it is reasonable to compare the estimates using their asymptotic variances. Thus the efficiency of one estimate relative to another is defined as the ratio of their asymptotic variances, and these efficiencies are evaluated numerically and analytically for a variety of nonnormal situations. Various authors, including Hocking and Kutner (1975), Corbeil and Searle (1976), and Swallow and Monahan (1984), have compared the variances and/or mean squared errors of variance component estimates in a variety of finite-sample models assuming normally distributed effects. The purpose of this article is to extend this line of research to incorporate nonnormal distributions and large samples. Some special cases of MINQUE considered in this article are called MINQUE(0), MINQUE(1), MINQUE(∞), and MIVQUE (minimum variance quadratic unbiased estimator); these are defined by setting the a priori variance ratio to 0, 1, infinity (in a sense to be explained), and to the actual variance ratio, respectively. Because the actual variance ratio is usually unknown, the MIVQUE estimates cannot be computed for unbalanced designs. (In balanced designs MIVQUE and ANOVA correspond.) It turns out, however, that ML and REML are asymptotically equivalent to MIVQUE, implying that only the asymptotic variance of MIVQUE is needed for evaluating efficiencies relative to ML or REML. Finite-sample efficiencies certainly may differ from the asymptotic counterparts, and the theoretical results are supplemented with a modest simulation study. This study indicates that the asymptotic results are reasonably close in moderate sample sizes for a particular type of design. The MIVQUE estimates have minimum variance within the class of invariant quadratic unbiased estimators under normality, but it is demonstrated that other commonly used estimates may be more efficient in nonnormal situations. The potential gain in efficiency, however, is often small relative to the potential loss of efficiency relative to MIVQUE. These results imply that the REML and ML estimates, whose derivations depend on the assumption that the data follow a multivariate normal distribution, are good choices in some nonnormal situations. Analysis of the large-sample variance shows that certain comparisons are invariant to the underlying distributions. For example, the efficiency of the ANOVA to MIVQUE estimates of between-classes variance becomes independent of the kurtosis parameters at the extreme values of the variance ratio, implying that REML and ML also dominate ANOVA (at least asymptotically) at these extremes. As the variance ratio tends to infinity, it may also be demonstrated that the efficiency of the MINQUE(0) estimate of the error variance (relative to ANOVA and MIVQUE) tends to 0 for arbitrary underlying distributions. This result supplements the simulation studies of Swallow and Monahan (1984), who observed similar behavior in normally distributed models. The phrase arbitrary underlying distributions is used throughout the article, but it is tacitly assumed that certain moment constraints must be satisfied.

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