Precise Asymptotics for Linear Mixed Models with Crossed Random Effects

TL;DR

This paper derives precise asymptotic normality results for maximum likelihood estimators in a broad class of Gaussian linear mixed models with crossed random effects, addressing complex dependencies and unbalanced data.

Contribution

It introduces a new theoretical framework for asymptotic analysis of crossed random effects models, extending beyond nested structures and balanced data.

Findings

Asymptotic normality of estimators established

Handles unbalanced and complex crossed random effects

Provides tools for confidence intervals and hypothesis testing

Abstract

We obtain an asymptotic normality result that reveals the precise asymptotic behavior of the maximum likelihood estimators of parameters for a very general class of linear mixed models containing cross random effects. In achieving the result, we overcome theoretical difficulties that arise from random effects being crossed as opposed to the simpler nested random effects case. Our new theory is for a class of Gaussian response linear mixed models which includes crossed random slopes that partner arbitrary multivariate predictor effects and does not require the cell counts to be balanced. Statistical utilities include confidence interval construction, Wald hypothesis test and sample size calculations.