On the estimation of the extremal index based on scaling and resampling

TL;DR

This paper introduces a new estimator for the extremal index in stationary time series, based on scaling and resampling, demonstrating its consistency, normality, and practical utility through simulations and real data applications.

Contribution

A novel extremal index estimator using asymptotic scaling and resampling, with automatic tuning and confidence intervals, validated through simulations and real-world data.

Findings

Estimator is consistent and asymptotically normal for m-dependent series.

Simulation results show highly competitive performance.

Applications to real data demonstrate diagnostic capabilities.

Abstract

The extremal index parameter theta characterizes the degree of local dependence in the extremes of a stationary time series and has important applications in a number of areas, such as hydrology, telecommunications, finance and environmental studies. In this study, a novel estimator for theta based on the asymptotic scaling of block-maxima and resampling is introduced. It is shown to be consistent and asymptotically normal for a large class of m-dependent time series. Further, a procedure for the automatic selection of its tuning parameter is developed and different types of confidence intervals that prove useful in practice proposed. The performance of the estimator is examined through simulations, which show its highly competitive behavior. Finally, the estimator is applied to three real data sets of daily crude oil prices, daily returns of the S&P 500 stock index, and high-frequency, intra-day traded volumes of a stock. These applications demonstrate additional diagnostic features of statistical plots based on the new estimator.