Asymptotics for testing hypothesis in some multivariate variance components model under non-normality

Abstract

We consider the problem of deriving the asymptotic distribution of the three commonly used multivariate test statistics, namely likelihood ratio, Lawley–Hotelling and Bartlett–Nanda–Pillai statistics, for testing hypotheses on the various effects (main, nested or interaction) in multivariate mixed models. We derive the distributions of these statistics, both in the null as well as non-null cases, as the number of levels of one of the main effects (random or fixed) goes to infinity. The robustness of these statistics against departure from normality will be assessed. Essentially, in the asymptotic spirit of this paper, both the hypothesis and error degrees of freedom tend to infinity at a fixed rate. It is intuitively appealing to consider asymptotics of this type because, for example, in random or mixed effects models, the levels of the main random factors are assumed to be a random sample from a large population of levels. For the asymptotic results of this paper to hold, we do not require any distributional assumption on the errors. That means the results can be used in real-life applications where normality assumption is not tenable. As it happens, the asymptotic distributions of the three statistics are normal. The statistics have been found to be asymptotically null robust against the departure from normality in the balanced designs. The expressions for the asymptotic means and variances are fairly simple. That makes the results an attractive alternative to the standard asymptotic results. These statements are favorably supported by the numerical results.