We investigate the estimation of multivariate extreme models with a discrete spectral measure using spherical clustering techniques. The primary contribution involves devising a method for selecting the order, that is, the number of clusters. The method consistently identifies the true order, i.e., the number of spectral atoms, and enjoys intuitive implementation in practice. Specifically, we introduce an extra penalty term to the well-known simplified average silhouette width, which penalizes small cluster sizes and small dissimilarities between cluster centers. Consequently, we provide a consistent method for determining the order of a max-linear factor model, where a typical information-based approach is not viable. Our second contribution is a large-deviation-type analysis for estimating the discrete spectral measure through clustering methods, which serves as an assessment of the convergence quality of clustering-based estimation for multivariate extremes. Additionally, as a third contribution, we discuss how estimating the discrete measure can lead to parameter estimations of heavy-tailed factor models. We also present simulations and real-data studies that demonstrate order selection and factor model estimation.

翻译：本文研究利用球面聚类技术估计具有离散谱测度的多元极值模型。主要贡献在于设计了一种选择阶数（即聚类数量）的方法。该方法能够一致识别真实阶数（即谱原子数量），且在实际应用中具有直观的可操作性。具体而言，我们在经典的简化平均轮廓宽度准则中引入了一个额外的惩罚项，该惩罚项针对小规模聚类与聚类中心间微小差异进行约束。由此，我们为最大线性因子模型提供了一种一致的阶数确定方法，而传统的基于信息准则的方法在此类模型中往往不可行。第二个贡献是通过聚类方法估计离散谱测度的大偏差型分析，该分析可用于评估基于聚类的多元极值估计的收敛质量。此外，作为第三个贡献，我们探讨了如何通过离散测度估计推导重尾因子模型的参数估计。文中还通过模拟实验与真实数据研究展示了阶数选择与因子模型估计的实际应用效果。