Variance Components Testing in ANOVA-Type Mixed Models

The purpose of this article is threefold. First, variance components testing for ANOVA-type mixed models is considered, in which response may not be divided into independent sub-vectors, whereas most of existing methods are for models where response can be divided into independent sub-vectors. Second, testing that a certain subset of variance components is zero. Third, as normality is often violated in practice, it is desirable to construct tests under very mild assumptions. To achieve these goals, an adaptive difference-based test and an adaptive trace-based test are constructed. The test statistics are asymptotically normal under the null hypothesis, are consistent against all global alternatives and can detect local alternatives distinct from the null at a rate as close to n^-1/2 as possible with n being the sample size. Moreover, when the dimensions of variance components in different sets are bounded, we develop a test with chi-square as its limiting null distribution. The finite sample performance of the tests is examined via simulations, and a real data set is analysed for illustration.