Inference for the limiting cluster size distribution of extreme values

TL;DR

This paper introduces recursive estimators for the limiting cluster size distribution and extremal index in stationary sequences, providing theoretical analysis and finite sample evaluations for understanding extreme value clustering.

Contribution

It presents novel recursive estimators for cluster size probabilities and extremal index, with proven asymptotic properties and simulation-based performance assessment.

Findings

Estimators are consistent and asymptotically normal.

Finite sample simulations show good estimator performance.

Method effectively captures cluster size distribution in extremes.

Abstract

Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound Poisson under appropriate long range dependence conditions. Typically exceedances appear in clusters. The underlying Poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes. In the present paper we introduce estimators of the limiting cluster size probabilities, which are constructed through a recursive algorithm. We derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions. We study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data.