A penalized least squares estimator for extreme-value mixture models

TL;DR

This paper introduces a penalized least squares estimator for max-stable models that effectively identifies parameters at the boundary and detects groups of variables experiencing simultaneous extremes.

Contribution

It proposes a novel pseudo-norm penalization approach with a data-driven algorithm to estimate parameters and identify extreme directions in mixture models.

Findings

Estimator performs well in simulations for parameter accuracy.

Method successfully identifies groups of variables with simultaneous extremes.

Applied to river discharge and stock loss data with meaningful results.

Abstract

Estimating the parameters of max-stable parametric models poses significant challenges, particularly when some parameters lie on the boundary of the parameter space. This situation arises when a subset of variables exhibits extreme values simultaneously, while the remaining variables do not -- a phenomenon commonly referred to as an extreme direction. A novel estimator is proposed for the parameters of a general parametric mixture model, incorporating a threshold exceedances approach based on a pseudo-norm penalization. The latter plays a crucial role in accurately identifying parameters at the boundary of the parameter space. Additionally, the estimator comes with a data-driven algorithm to detect groups of variables corresponding to extreme directions. The performance of the estimator is assessed in terms of both parameter estimation and the identification of extreme directions through extensive simulation studies. Finally, the method is applied to two real-world datasets: discharge measurements at stations along the Danube river, and financial portfolio losses from stocks listed on the NYSE, AMEX, and NASDAQ. In both applications, the sets of variables that can become large simultaneously are identified.