Over the past decades, linear mixed models have attracted considerable attention in various fields of applied statistics. They are popular whenever clustered, hierarchical or longitudinal data are investigated. Nonetheless, statistical tools for valid simultaneous inference for mixed parameters are rare. This is surprising because one often faces inferential problems beyond the pointwise examination of fixed or mixed parameters. For example, there is an interest in a comparative analysis of cluster-level parameters or subject-specific estimates in studies with repeated measurements. We discuss methods for simultaneous inference assuming a linear mixed model. Specifically, we develop simultaneous prediction intervals as well as multiple testing procedures for mixed parameters. They are useful for joint considerations or comparisons of cluster-level parameters. We employ a consistent bootstrap approximation of the distribution of max-type statistic to construct our tools. The numerical performance of the developed methodology is studied in simulation experiments and illustrated in a data example on household incomes in small areas.
Linear mixed effects are considered excellent predictors of cluster-level parameters in various domains. However, previous work has shown that their performance can be seriously affected by departures from modelling assumptions. Since the latter are common in applied studies, there is a need for inferential methods which are to certain extent robust to misspecfications, but at the same time simple enough to be appealing for practitioners. We construct statistical tools for cluster-wise and simultaneous inference for mixed effects under model misspecification using straightforward semiparametric random effect bootstrap. In our theoretical analysis, we show that our methods are asymptotically consistent under general regularity conditions. In simulations our intervals were robust to severe departures from model assumptions and performed better than their competitors in terms of empirical coverage probability.
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