We consider a linear mixed-effects model with a clustered structure, where the parameters are estimated using maximum likelihood (ML) based on possibly unbalanced data. Inference with this model is typically done based on asymptotic theory, assuming that the number of clusters tends to infinity with the sample size. However, when the number of clusters is fixed, classical asymptotic theory developed under a divergent number of clusters is no longer valid and can lead to erroneous conclusions. In this paper, we establish the asymptotic properties of the ML estimators of random-effects parameters under a general setting, which can be applied to conduct valid statistical inference with fixed numbers of clusters. Our asymptotic theorems allow both fixed effects and random effects to be misspecified, and the dimensions of both effects to go to infinity with the sample size.

我们考虑具有聚类结构的线性混合效应模型，其中参数是基于可能不平衡的数据使用最大似然 (ML) 估计的。该模型的推断通常是基于渐近理论进行的，假设集群的数量随着样本量趋于无穷大。然而，当集群的数量固定时，在不同的集群数量下发展起来的经典渐近理论不再有效，并且可能导致错误的结论。在本文中，我们建立了在一般设置下随机效应参数的 ML 估计量的渐近性质，可用于在固定数量的聚类下进行有效的统计推断。我们的渐近定理允许错误指定固定效应和随机效应，