We propose new methods for multivariate linear regression when the regression coefficient matrix is sparse and the error covariance matrix is dense. We assume that the error covariance matrix has equicorrelation across the response variables. Two procedures are proposed: one is based on constant marginal response variance (compound symmetry), and the other is based on general varying marginal response variance. Two approximate procedures are also developed for high dimensions. We propose an approximation to the Gaussian validation likelihood for tuning parameter selection. Extensive numerical experiments illustrate when our procedures outperform relevant competitors as well as their robustness to model misspecification.

翻译：本文针对回归系数矩阵稀疏而误差协方差矩阵稠密的多元线性回归问题，提出了新的方法。我们假设误差协方差矩阵在响应变量间具有等相关性。提出了两种处理方案：一种基于恒定的边际响应方差（复合对称性），另一种基于一般性变化的边际响应方差。针对高维场景，我们还开发了两种近似处理流程。为进行调参选择，我们提出了高斯验证似然函数的近似计算方法。大量数值实验表明，我们的方法在特定条件下优于相关竞争方法，并对模型设定误差具有较好的鲁棒性。