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The Discrepancy Method for Extremal Index Estimation

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Nonparametric Statistics (ISNPS 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 339))

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Abstract

We consider the nonparametric estimation of the extremal index of stochastic processes. The discrepancy method that was proposed by the author as a data-driven smoothing tool for probability density function estimation is extended to find a threshold parameter u for an extremal index estimator in case of heavy-tailed distributions. To this end, the discrepancy statistics are based on the von Mises–Smirnov statistic and the k largest order statistics instead of an entire sample. The asymptotic chi-squared distribution of the discrepancy measure is derived. Its quantiles may be used as discrepancy values. An algorithm to select u for an estimator of the extremal index is proposed. The accuracy of the discrepancy method is checked by a simulation study.

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Notes

  1. 1.

    \(L=1\) holds when \(\theta =0\).

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Acknowledgments

The author appreciates the partial financial support by the Russian Foundation for Basic Research, grant 19-01-00090.

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Authors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, 117997, Russia

    Natalia Markovich

Authors
  1. Natalia Markovich
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Correspondence to Natalia Markovich .

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Editors and Affiliations

  1. Department of Economics and Statistics, University of Salerno, Salerno, Italy

    Michele La Rocca

  2. Department of Methods and Models for Economics, Terrority and Finance, Sapienza University of Rome, Rome, Italy

    Brunero Liseo

  3. Department of Management and Engineering, University of Padua, Vicenza, Italy

    Luigi Salmaso

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Markovich, N. (2020). The Discrepancy Method for Extremal Index Estimation. In: La Rocca, M., Liseo, B., Salmaso, L. (eds) Nonparametric Statistics. ISNPS 2018. Springer Proceedings in Mathematics & Statistics, vol 339. Springer, Cham. https://doi.org/10.1007/978-3-030-57306-5_31

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