  We consider three problems in high-dimensional Gaussian linear mixed models.
Without any assumptions on the design for the fixed effects, we construct an
asymptotic $F$-statistic for testing whether a collection of random effects is
zero, derive an asymptotic confidence interval for a single random effect at
the parametric rate $\sqrt{n}$, and propose an empirical Bayes estimator for a
part of the mean vector in ANOVA type models that performs asymptotically as
well as the oracle Bayes estimator. We support our results with numerical
simulations and provide comparisons with oracle estimators. The procedures
developed are applied to the Trends in International Mathematics and Sciences
Study (TIMSS) data.
