This paper addresses a long-standing open problem in the analysis of linear mixed models with crossed random effects under unbalanced designs: how to find an analytic expression for the inverse of $\mathbf{V}$, the covariance matrix of the observed response. The inverse matrix $\mathbf{V}^{-1}$ is required for likelihood-based estimation and inference. However, for unbalanced crossed designs, $\mathbf{V}$ is dense and the lack of a closed-form representation for $\mathbf{V}^{-1}$, until now, has made using likelihood-based methods computationally challenging and difficult to analyse mathematically. We use the Khatri--Rao product to represent $\mathbf{V}$ and then to construct a modified covariance matrix whose inverse admits an exact spectral decomposition. Building on this construction, we obtain an elegant and simple approximation to $\mathbf{V}^{-1}$ for asymptotic unbalanced designs. For non-asymptotic settings, we derive an accurate and interpretable approximation under mildly unbalanced data and establish an exact inverse representation as a low-rank correction to this approximation, applicable to arbitrary degrees of unbalance. Simulation studies demonstrate the accuracy, stability, and computational tractability of the proposed framework.

翻译：本文解决了线性混合模型中非平衡交叉随机效应设计下一个长期存在的开放性问题：如何找到观测响应协方差矩阵$\mathbf{V}$的逆矩阵的解析表达式。基于似然的估计与推断需要逆矩阵$\mathbf{V}^{-1}$。然而，对于非平衡交叉设计，$\mathbf{V}$是稠密矩阵，且迄今为止缺乏$\mathbf{V}^{-1}$的闭式表示，这使得基于似然的方法在计算上具有挑战性且数学分析困难。我们利用Khatri–Rao积表示$\mathbf{V}$，进而构造一个修正的协方差矩阵，其逆矩阵允许精确的谱分解。基于此构造，我们得到了渐近非平衡设计下$\mathbf{V}^{-1}$的一个优雅且简洁的近似。对于非渐近情形，我们在轻度非平衡数据下推导出一个精确且可解释的近似，并建立了精确逆矩阵的表示形式，即作为该近似的低秩修正，适用于任意程度的非平衡。仿真研究验证了所提框架的精确性、稳定性和计算可行性。