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A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions

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  • Pierre Lafaye de Micheaux

    (School of Mathematics and Statistics, UNSW Sydney, Sydney, NSW 2052, Australia
    Desbrest Institute of Epidemiology and Public Health, INSERM and University Montpellier, 34093 Montpellier, France
    AMIS, Université Paul Valéry Montpellier 3, 34199 Montpellier, France)

  • Frédéric Ouimet

    (Department of Mathematics and Statistics, McGill University, Montreal, QC H3A 0B9, Canada
    Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA)

Abstract

In this paper, we complement a study recently conducted in a paper of H.A. Mombeni, B. Masouri and M.R. Akhoond by introducing five new asymmetric kernel c.d.f. estimators on the half-line [ 0 , ∞ ) , namely the Gamma, inverse Gamma, LogNormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum–Saunders and Weibull kernel c.d.f. estimators from Mombeni, Masouri and Akhoond. By using the same experimental design, we show that the LogNormal and Birnbaum–Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method from C. Tenreiro.

Suggested Citation

  • Pierre Lafaye de Micheaux & Frédéric Ouimet, 2021. "A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions," Mathematics, MDPI, vol. 9(20), pages 1-35, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:20:p:2605-:d:657743
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    References listed on IDEAS

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