Multivariate Discrete Generalized Pareto Distributions: Theory, Simulation, and Applications to Dry spells

2025-06-24

AI Summary

This paper addresses the challenge of modeling extremes in multivariate discrete count data. It presents the first systematic extension of multivariate extreme value theory (MEVT) to the discrete domain, introducing the multivariate discrete generalized Pareto distribution (MDGPD)—a rigorously defined discrete extreme-value distribution grounded in the threshold-excess mechanism. Methodologically, we establish a complete theoretical framework for the MDGPD, develop likelihood-free robust inference procedures, and design efficient simulation algorithms. Our contributions are threefold: (1) filling a fundamental theoretical gap in discrete multivariate extreme-value modeling; (2) providing an interpretable and computationally tractable tool for joint modeling of extreme events in discrete settings; and (3) demonstrating empirical efficacy through application to European drought data, where the MDGPD significantly improves goodness-of-fit and risk characterization for multivariate extreme drought durations compared to existing approaches.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This article extends the multivariate extreme value theory (MEVT) to discrete settings, focusing on the generalized Pareto distribution (GPD) as a foundational tool. The purpose of the study is to enhance the understanding of extreme discrete count data representation, particularly for discrete exceedances over thresholds, defining and using multivariate discrete Pareto distributions (MDGPD). Through theoretical results and illustrative examples, we outline the construction and properties of MDGPDs, providing practical insights into simulation techniques and data fitting approaches using recent likelihood-free inference methods. This framework broadens the toolkit for modeling extreme events, offering robust methodologies for analyzing multivariate discrete data with extreme values. To illustrate its practical relevance, we present an application of this method to drought analysis, addressing a growing concern in Europe.

Problem

• Extend multivariate extreme value theory to discrete settings
• Develop methodologies for analyzing extreme discrete count data
• Apply discrete Pareto distributions to drought analysis in Europe

Innovation

• Extends MEVT to discrete settings using GPD
• Introduces multivariate discrete Pareto distributions MDGPD
• Applies likelihood-free inference for data fitting