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.

翻译：本文推导了交叉混合效应模型中个体及单元水平随机效应的经验最佳线性无偏预测（EBLUPs）的联合渐近分布。在温和条件下（包括随机效应和模型误差的矩条件而非正态性假设），我们证明当行、列以及包含交互作用时的单元规模同时趋于无穷时，EBLUPs与随机效应之差的分布满足中心极限定理。这些中心极限定理意味着EBLUPs渐近服从真实随机效应分布与正态分布的卷积。此外，我们的研究结果能够为EBLUPs的均方误差（MSE）提供简单的渐近近似与估计量，从而有助于构建未观测随机效应的渐近预测区间。仿真实验表明，我们提出的EBLUPs均方误差简单估计量在有限样本中表现优异。最后，我们通过电影评分数据分析展示了渐近预测区间的应用。