Asymptotic Normality of the Anova Estimates of Components of Variance in the Nonnormal, Unbalanced Hierarchal Mixed Model

TLDR: In this article, the authors considered the non-normal, unbalanced hierarchal design and mild conditions for a sequence of such designs are specified so that the vector of normalized ANOVA estimates converges to a multivariate normal distribution.

Abstract: Despite their lack of optimality in unbalanced normally distributed models, the ANOVA estimates of components of variance are convenient and widely used. The hierarchal (nested) design is well suited to this estimation scheme. In this paper the nonnormal, unbalanced hierarchal design is considered and mild conditions for a sequence of such designs are specified so that the vector of normalized ANOVA estimates converges to a multivariate normal distribution. The nested structure permits an expression of the estimates in terms of a sum of independent quadratic forms in mean zero random variables plus smaller order remainder, and a theorem of Whittle (1960) establishes the Liapounov criterion. Distinguishing features of this paper are the limit theory of nonnormal unbalanced models and the allowance that some variances other than the error variance may be null.