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Nonparametric Estimation of Heavy-Tailed Density by the Discrepancy Method

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Nonparametric Statistics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 175))

Abstract

The nonparametric estimation of the probability density function (pdf) requires smoothing parameters like bandwidths of kernel estimates. We consider the so-called discrepancy method proposed in [13, 14, 21] as a data-driven smoothing tool and alternative to cross-validation. It is based on the von Mises-Smirnov’s (M-S) and the Kolmogorov–Smirnov’s (K-S) nonparametric statistics as measures in the space of distribution functions (cdfs). The unknown smoothing parameter is found as a solution of the discrepancy equation. On its left-hand side stands the measure between the empirical distribution function and the nonparametric estimate of the cdf. The latter is obtained as a corresponding integral of the pdf estimator. The right-hand side is equal to a quantile of the asymptotic distribution of the M-S or K-S statistic. The discrepancy method considered earlier for light-tailed pdfs is investigated now for heavy-tailed pdfs.

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Notes

  1. 1.

    For example, the Hill’s estimator of the tail index is based on the k largest statistics.

  2. 2.

    [x] denotes the integer part of x.

  3. 3.

    The rule-of-thumb selected \(h_2\) is recommended in ([20], p. 45) as an optimal value for the Gaussian kernel. This method is however very sensitive to outliers due to possibly large \(\sigma \).

  4. 4.

    This is ksdensity-procedure in Matlab.

  5. 5.

    It was calculated by Dynara Team Matlab code \(mh-optimal-bandwidth\).

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Acknowledgments

This work was supported in part by the Russian Foundation for Basic Research, grant 13-08-00744.

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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

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  1. Natalia Markovich
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Correspondence to Natalia Markovich .

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Editors and Affiliations

  1. Department of Mathematics, Universidade da Coruña, A Coruña, Spain

    Ricardo Cao

  2. Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain

    Wenceslao González Manteiga

  3. Department of Statistics, Universidad Carlos III de Madrid, Getafe, Spain

    Juan Romo

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Markovich, N. (2016). Nonparametric Estimation of Heavy-Tailed Density by the Discrepancy Method. In: Cao, R., González Manteiga, W., Romo, J. (eds) Nonparametric Statistics. Springer Proceedings in Mathematics & Statistics, vol 175. Springer, Cham. https://doi.org/10.1007/978-3-319-41582-6_8

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