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On the Multivariate Extremal Index.

S Nandagopalan1

  • 1Colorado State University, Fort Collins, CO 80523.

Journal of Research of the National Institute of Standards and Technology
|July 5, 2023
PubMed
Summary
This summary is machine-generated.

This study extends the exceedance point process approach to multivariate stationary sequences, revealing the impact of clustering on limiting behavior and linking point process convergence to maxima. A new multivariate extremal index is introduced with properties similar to its univariate version.

Keywords:
dependence functionexceedanceextremal indexmultivariatepoint processstationary

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Area of Science:

  • Probability Theory
  • Extreme Value Theory
  • Stochastic Processes

Background:

  • High-level exceedances in stationary sequences often exhibit a Compound Poisson structure due to clustering.
  • Understanding this clustering is crucial for accurate modeling of extreme events.
  • Existing methods primarily focus on univariate cases.

Purpose of the Study:

  • To extend the exceedance point process approach to multivariate stationary sequences.
  • To investigate the precise effect of clustering on limiting distributions.
  • To explore the relationship between point process convergence and the behavior of maxima.

Main Methods:

  • Extension of Hsing et al.'s exceedance point process approach.
  • Analysis of weak convergence for multivariate stationary sequences.
  • Introduction and characterization of the multivariate extremal index.

Main Results:

  • Weak convergence results are obtained for multivariate stationary sequences.
  • The precise effect of clustering on limiting distributions is clarified.
  • The multivariate extremal index is shown to have properties analogous to the univariate case.
  • Calculations of the extremal index for specific bivariate moving average sequences are presented.

Conclusions:

  • The study successfully extends the exceedance point process framework to multivariate settings.
  • The introduced multivariate extremal index provides a valuable tool for analyzing extreme events in multivariate data.
  • The findings contribute to a deeper understanding of extreme value theory in complex stochastic systems.