Threshold selection for extremal index estimation

TL;DR

This paper introduces a novel data-driven threshold selection method for extremal index estimation using a modified discrepancy approach based on the Cramér-von Mises-Smirnov statistic, improving automatic selection in stochastic process analysis.

Contribution

It adapts the discrepancy method for threshold selection in extremal index estimation, providing asymptotic distribution and convergence rate analysis.

Findings

The method effectively selects thresholds for extremal index estimators.

Asymptotic distribution of the discrepancy statistic matches the classical omega-squared distribution.

The approach can be applied to various extremal index estimators for improved automatic threshold choice.

Abstract

We propose a new threshold selection method for the nonparametric estimation of the extremal index of stochastic processes. The so-called discrepancy method was proposed as a data-driven smoothing tool for estimation of a probability density function. Now it is modified to select a threshold parameter of an extremal index estimator. To this end, a specific normalization of the discrepancy statistic based on the Cram\'{e}r-von Mises-Smirnov statistic $\omega^2$ is calculated by the $k$ largest order statistics instead of an entire sample. Its asymptotic distribution as $k\to\infty$ is proved to be the same as the $\omega^2$-distribution. The quantiles of the latter distribution are used as discrepancy values. The rate of convergence of an extremal index estimate coupled with the discrepancy method is derived. The discrepancy method is used as an automatic threshold selection for the intervals and $K-$gaps estimators and it may be applied to other estimators of the extremal index.