Increasing dimension asymptotics for two-way crossed mixed effect models

Abstract

This paper presents asymptotic results for the maximum likelihood and restricted maximum likelihood (REML) estimators within a two-way crossed mixed effect model, when the number of rows, columns, and the number of observations per cell tend to infinity. The relative growth rate for the number of rows, columns, and cells is unrestricted, whether considered pairwise or collectively. Under very mild conditions (which include moment conditions instead of requiring normality for either the random effects or errors), the estimators are proven to be asymptotically normal, with a structured covariance matrix. We also discuss the case where the number of observations per cell is fixed at 1.