Estimating the Upcrossings Index

TL;DR

This paper introduces a new estimator for the upcrossings index in stationary sequences, demonstrating its consistency and asymptotic normality, with validation through simulations and real-world case studies in environment and finance.

Contribution

It proposes a novel estimator for the upcrossings index under mild conditions, extending the tools for analyzing clustering of high levels in stationary sequences.

Findings

Estimator is consistent and asymptotically normal.

Simulation studies show good performance for autoregressive processes.

Case studies illustrate practical applicability in environment and finance.

Abstract

For stationary sequences, under general local and asymptotic dependence restrictions, any limiting point process for time normalized upcrossings of high levels is a compound Poisson process, i.e., there is a clustering of high upcrossings, where the underlying Poisson points represent cluster positions, and the multiplicities correspond to cluster sizes. For such classes of stationary sequences there exists the upcrossings index $\eta,$ $0\leq \eta\leq 1,$ which is directly related to the extremal index $\theta,$ $0\leq \theta\leq 1,$ for suitable high levels. In this paper we consider the problem of estimating the upcrossings index $\eta$ for a class of stationary sequences satisfying a mild oscillation restriction. For the proposed estimator, properties such as consistency and asymptotic normality are studied. Finally, the performance of the estimator is assessed through simulation studies for autoregressive processes and case studies in the fields of environment and finance.