This paper studies Graphical SLOPE for precision matrix estimation, with emphasis on its ability to recover both sparsity and clusters of edges with equal or similar strength. In a fixed-dimensional regime, we establish that the root-$n$ scaled estimation error converges to the unique minimizer of a strictly convex optimization problem defined through the directional derivative of the SLOPE penalty. We also establish convergence of the induced SLOPE pattern, thereby obtaining an asymptotic characterization of the clustering structure selected by the estimator. A comparison with GLASSO shows that the grouping property of SLOPE can substantially improve estimation accuracy when the precision matrix exhibits structured edge patterns. To assess the effect of departures from Gaussianity, we then analyze Gaussian-loss precision matrix estimation under elliptical distributions. In this setting, we derive the limiting distribution and quantify the inflation in variability induced by heavy tails relative to the Gaussian benchmark. We also study TSLOPE, based on the multivariate $t$-loss, and derive its limiting distribution. The results show that TSLOPE offers clear advantages over GSLOPE under heavy-tailed data-generating mechanisms. Simulation evidence suggests that these qualitative conclusions persist in high-dimensional settings, and an empirical application shows that SLOPE-based estimators, especially TSLOPE, can uncover economically meaningful clustered dependence structures.

翻译：本文研究用于精度矩阵估计的图形化SLOPE方法，重点探讨其恢复稀疏性以及具有相等或相似强度的边聚类的能⼒。在固定维度场景下，我们证明了根号n缩放后的估计误差收敛于通过SLOPE惩罚的方向导数定义的严格凸优化问题的唯一最小化解。我们还建立了诱导SLOPE模式的收敛性，从而获得估计器所选择的聚类结构的渐近特征描述。与GLASSO的比较表明，当精度矩阵呈现结构化边模式时，SLOPE的分组特性可以显著提升估计精度。为评估偏离高斯性的影响，我们随后分析了椭圆分布下高斯损失精度矩阵估计的性态。在此设定下，我们推导了极限分布，并量化了重尾分布相对于高斯基准所导致的变异性膨胀。同时，基于多元t损失研究了TSLOPE，并推导了其极限分布。结果表明，在重尾数据生成机制下，TSLOPE相较于GSLOPE具有明显优势。模拟证据表明这些定性结论在高维设定下仍成立，而实证应用显示基于SLOPE的估计器（尤其是TSLOPE）能够揭示具有经济意义的聚类依赖结构。