Asymptotics for EBLUPs within crossed mixed effect models

TL;DR

This paper derives the asymptotic distribution of EBLUPs in crossed mixed effect models, enabling improved inference and prediction intervals under mild conditions.

Contribution

It provides the first joint asymptotic distribution results for EBLUPs in crossed models, including simple MSE estimators and practical prediction intervals.

Findings

Asymptotic normality of EBLUP differences established

Effective MSE estimators demonstrated in simulations

Application to movie rating data illustrates practical utility

Abstract

In this article, we derive the joint asymptotic distribution of empirical best linear unbiased predictors (EBLUPs) for individual and cell-level random effects in a crossed mixed effect model. Under mild conditions (which include moment conditions instead of normality for the random effects and model errors), we demonstrate that as the sizes of rows, columns, and, when we include interactions, cells simultaneously increase to infinity, the distribution of the differences between the EBLUPs and the random effects satisfy central limit theorems. These central limit theorems mean the EBLUPs asymptotically follow the convolution of the true random effect distribution and a normal distribution. Moreover, our results enable simple asymptotic approximations and estimators for the mean squared error (MSE) of the EBLUPs, which in turn facilitates the construction of asymptotic prediction intervals for the unobserved random effects. We show in simulations that our simple estimator of the MSE of the EBLUPs works very well in finite samples. Finally, we illustrate the use of the asymptotic prediction intervals with an analysis of movie rating data.