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

## TL;DR

This paper develops methods for valid simultaneous inference on parameters in linear mixed models, enabling joint analysis and comparisons of cluster-level estimates, with demonstrated effectiveness through simulations and a real data example.

## Contribution

It introduces novel simultaneous prediction intervals and multiple testing procedures for mixed parameters in linear mixed models, using bootstrap methods.

## Key findings

- Methods perform well in simulations
- Effective in small area income estimation
- Allows joint inference on cluster-level parameters

## 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.