{"ID":2830525,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.09273","arxiv_id":"2512.09273","title":"On the inverse of covariance matrices for unbalanced crossed designs","abstract":"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.","short_abstract":"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 likeliho...","url_abs":"https://arxiv.org/abs/2512.09273","url_pdf":"https://arxiv.org/pdf/2512.09273v2","authors":"[\"Ziyang Lyu\",\"S. A. Sisson\",\"A. H. Welsh\"]","published":"2025-12-10T02:54:18Z","proceeding":"stat.ME","tasks":"[\"stat.ME\",\"math.ST\"]","methods":"[\"Generative Adversarial Network\"]","has_code":false}