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EXTREME VALUE THEORY WITH OPERATOR NORMING.

Mark M Meerschaert1, Hans-Peter Scheffler2, Stilian A Stoev3

  • 1Department of Statistics & Probability, Michigan State University, East Lansing MI 48824 USA. URL: http://www.stt.msu.edu/users/mcubed/

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Summary
This summary is machine-generated.

This study introduces a novel method for analyzing extreme value theory in heavy-tailed vector data. It allows the tail index to vary directionally, offering a more nuanced understanding of extreme events.

Keywords:
directional extremesheavy tailshetero-ouracityoperator regular variationparametric bootstrapspectral representation

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Area of Science:

  • Statistics
  • Probability Theory
  • Data Analysis

Background:

  • Extreme value theory (EVT) traditionally assumes a constant tail index for heavy-tailed data.
  • Analyzing directional variations in tail behavior is crucial for understanding multivariate extremes.

Purpose of the Study:

  • To develop a new approach for EVT applicable to vector data with heavy tails.
  • To investigate and model directional variations in the tail index.

Main Methods:

  • Utilizing operator regular variation for theoretical development.
  • Employing extremal integrals to analyze asymptotic behavior.
  • Developing a statistical test to detect directional variation in the tail index.

Main Results:

  • A new theoretical framework for directional extreme value analysis is established.
  • The proposed test provides a method to empirically assess directional tail index variation.
  • Demonstrates that the tail index can vary with direction, not limited to coordinate axes.

Conclusions:

  • The developed approach offers a more flexible and accurate method for extreme value analysis in high-dimensional data.
  • The directional tail index variation is a significant feature in certain heavy-tailed vector datasets.
  • The proposed test is a valuable tool for applied statisticians and data scientists working with extreme events.