U-tests for variance components in linear mixed models

We propose U -statistics-based tests for null variance components in linear mixed models. We obtain their asymptotic distribution (for increasing number of units) under mild regularity conditions that include only the existence of the second moment for the distribution of the random effects and of the fourth moment for the distribution of the conditional errors. We also derive their nonnull distributions under local alternatives under the additional assumption of the existence of the fourth moment for the distribution of the random effects. Our proposal is easy to implement and may be applied to a wide class of linear mixed models. We also consider a simulation study to evaluate the behaviour of the U -test in small and moderate samples and compare its performance with that of exact F -tests and of generalized likelihood ratio tests obtained under the assumption of normality. An practical example in which the normality assumption is not reasonable is included as illustration.