# Estimating the Random Effect in Big Data Mixed Models

## TL;DR

This paper develops new statistical methods for high-dimensional Gaussian linear mixed models, including tests, confidence intervals, and estimators for random effects, with applications to educational data analysis.

## Contribution

It introduces an asymptotic F-test, confidence interval, and empirical Bayes estimator for random effects in high-dimensional mixed models, without fixed effect design assumptions.

## Key findings

- The F-test asymptotically controls type I error.
- The confidence interval achieves parametric rate $\
- ,

## Abstract

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.