Statistical inference on $D^{(d)}(u_n)$ condition and estimation of the Extremal Index

TL;DR

This paper introduces a hypothesis test for the $D^{(d)}(u_n)$ dependence condition and develops an extremal index estimator, with theoretical properties and practical evaluation through simulations and a temperature data case study.

Contribution

It provides a new hypothesis test for the $D^{(d)}(u_n)$ condition and an extremal index estimator with proven asymptotic normality, enhancing extreme value analysis.

Findings

The hypothesis test effectively distinguishes models satisfying or violating $D^{(d)}(u_n)$.

The extremal index estimator is asymptotically normal.

Application to temperature data reveals clustering of extreme heat events.

Abstract

Clustering of extreme events can have profound and detrimental societal consequences. The extremal index, a number in the unit interval, is a key parameter in modelling the clustering of extremes. The study of extremal index often assumes a local dependence condition known as the $D^{(d)}(u_n)$ condition. In this paper, we develop a hypothesis test for $D^{(d)}(u_n)$ condition based on asymptotic results. We develop an estimator for the extremal index by leveraging the inference procedure based on the $D^{(d)}(u_n)$ condition, and we establish the asymptotic normality of this estimator. The finite sample performances of the hypothesis test and the estimation are examined in a simulation study, where we consider both models that satisfies the $D^{(d)}(u_n)$ condition and models that violate this condition. In a simple case study, our statistical procedure shows that daily temperature in summer shares a common clustering structure of extreme values based on the data observed in three weather stations in the Netherlands, Belgium and Spain.