  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.
