Statistics for Heteroscedastic Time Series Extremes

TL;DR

This paper extends a heteroscedastic extremes model for serially dependent time series, providing new estimators, limit theorems, and bootstrap tests, with validation through simulations.

Contribution

It introduces novel estimators and theoretical results for heteroscedastic extremes in dependent time series, including bootstrap testing and extremal index estimation.

Findings

Proved a local limit theorem for kernel estimators of the scedasis function.

Established a functional limit theorem for the integrated scedasis estimator.

Validated methods through Monte Carlo simulations.

Abstract

Einmahl, de Haan and Zhou (2016, Journal of the Royal Statistical Society: Series B, 78(1), 31-51) recently introduced a stochastic model that allows for heteroscedasticity of extremes. The model is extended to the situation where the observations are serially dependent, which is crucial for many practical applications. We prove a local limit theorem for a kernel estimator for the scedasis function, and a functional limit theorem for an estimator for the integrated scedasis function. We further prove consistency of a bootstrap scheme that allows to test for the null hypothesis that the extremes are homoscedastic. Finally, we propose an estimator for the extremal index governing the dynamics of the extremes and prove its consistency. All results are illustrated by Monte Carlo simulations. An important intermediate result concerns the sequential tail empirical process under serial dependence.