On estimation and order selection for multivariate extremes via clustering

TL;DR

This paper introduces a clustering-based method for estimating multivariate extreme models, including a novel approach for selecting the number of spectral clusters, with theoretical guarantees and practical validation through simulations and real data.

Contribution

It proposes a consistent clustering-based order selection method for multivariate extremes and analyzes its convergence properties, addressing limitations of traditional information-based approaches.

Findings

The method accurately identifies the true number of spectral atoms.

Clustering-based estimation converges reliably in large deviations analysis.

Application to real data demonstrates effective order and parameter estimation.

Abstract

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.