A New Regression Model for Multivariate Extremes Using the Peaks-Over-Threshold Method

Abstract

This paper presents a new regression model tailored for analyzing multivariate extremes, particularly when both the response variable and the covariates exhibit extreme behavior. The methodology combines the Peaks-Over-Threshold (POT) approach with ARIMA-based preprocessing to ensure the stationarity of climatic time series data before extreme value modeling. A key innovation of our approach is the use of a Logistic-Normal prior for spectral density estimation, which provides a more flexible and expressive alternative to the classical Dirichlet priors typically used in Peaks-Over-Threshold (POT) modeling of multivariate extremes. This framework enables the construction of smooth regression manifolds that describe conditional quantiles of extreme responses given extreme covariates, offering improved interpretability and adaptability to complex dependence structures. The model is applied to temperature and precipitation data from Meknes, Morocco (1973–2024), revealing meaningful extremal dependence patterns. Validation is conducted through quantile regression manifolds and bootstrap-based uncertainty estimates, demonstrating the practical relevance and robustness of the proposed method in environmental applications.

Keywords: Spectral measure; Bivariate extreme value distribution; Joint distribution; Quantile regression; Peaks-Over-Threshold; ARIMA method.

2010 Mathematics Subject Classification. 26A25; 26A35