On the inverse of covariance matrices for unbalanced crossed designs

Summary This paper addresses a long-standing open problem in crossed random effect models under unbalanced designs: how to find an analytic expression for the inverse of V, the covariance matrix of the observed response. For unbalanced crossed designs, V is dense and the lack of a closed-form representation for 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 V and then construct a modified covariance matrix whose inverse admits an exact spectral decomposition. Building on this construction, we obtain an elegant and simple approximation to 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. Simulations demonstrate the framework’s accuracy, stability, and tractability.