We obtain an asymptotic normality result that reveals the precise asymptotic behaviour 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 include crossed random slopes that partner arbitrary multivariate predictor effects and do not require the cell counts to be balanced. Statistical utilities include the confidence interval construction, Wald hypothesis test and sample size calculations.

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To cite this article: Jiming Jiang, Matt P. Wand & Swarnadip Ghosh (10 Mar 2026): Precise asymptotics for linear mixed models with crossed random effects, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2026.2633805

To link to this article: https://doi.org/10.1080/24754269.2026.2633805