Simultaneous inference for linear mixed model parameters with an application to small area estimation

Abstract

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.

A Note on Simultaneous Confidence Intervals for Direct, Indirect and Synthetic Estimators

“…For valid inference, we should then offer multiple tests or confidence intervals. Just recently, this was performed in the context of mixed model-based small area estimation [11,12,14]. To the best of our knowledge, we are the first who consider this problem for direct and indirect domain estimators which are still in frequent use.…”

“…Refs. [11,12] applied this idea to small area estimation and introduced bootstrap procedures to approximate the distribution of the resulting pivotal statistic. In our case, such a max-type statistic is…”

“…Recently, refs. [11,12] introduced different tools for simultaneous inference in the model-based small area estimation context, i.e., for estimators based on linear and generalized linear mixed models. For design-based estimators, we could not find any similar study; this paper is a contribution to fill this gap.…”