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具有交叉随机效应且设计不平衡情况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，对于不平衡交叉设计，$\mathbf{V}$ 是稠密的，且此前一直缺乏 $\mathbf{V}^{-1}$ 的闭式表示，这使得使用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 积来表示 $\mathbf{V}$，进而构造一个修正的协方差矩阵，其逆允许精确的谱分解。基于此构造，我们得到了渐近不平衡设计下 $\mathbf{V}^{-1}$ 的一个优雅且简单的近似。对于非渐近情形，我们在轻度不平衡数据下推导出一个准确且可解释的近似，并建立了一个精确的逆表示，作为对该近似的低秩修正，适用于任意程度的不平衡。模拟研究验证了所提出框架的准确性、稳定性和计算可行性。