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On the properties of hermite series based distribution function estimators

Author

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  • Michael Stephanou

    (University of Cape Town)

  • Melvin Varughese

    (University of Cape Town
    University of Western Australia)

Abstract

Hermite series based distribution function estimators have recently been applied in the context of sequential quantile estimation. These distribution function estimators are particularly useful because they allow the online (sequential) estimation of the full cumulative distribution function. This is in contrast to the empirical distribution function estimator and smooth kernel distribution function estimator which only allow sequential cumulative probability estimation at particular values on the support of the associated density function. Hermite series based distribution function estimators are well suited to the settings of streaming data, one-pass analysis of massive data sets and decentralised estimation. In this article we study these estimators in a more general context, thereby redressing a gap in the literature. In particular, we derive new asymptotic consistency results in the mean squared error, mean integrated squared error and almost sure sense. We also present novel Bias-robustness results for these estimators. Finally, we study the finite sample performance of the Hermite series based estimators through a real data example and simulation study. Our results indicate that in the general (non-sequential) context, the Hermite series based distribution function estimators are inferior to smooth kernel distribution function estimators, but may remain compelling in the context of sequential estimation of the full distribution function.

Suggested Citation

  • Michael Stephanou & Melvin Varughese, 2021. "On the properties of hermite series based distribution function estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(4), pages 535-559, May.
    • Handle: RePEc:spr:metrik:v:84:y:2021:i:4:d:10.1007_s00184-020-00785-z
    DOI: 10.1007/s00184-020-00785-z
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    References listed on IDEAS

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    1. Greblicki, Wlodzimierz & Pawlak, Miroslaw, 1985. "Pointwise consistency of the hermite series density estimate," Statistics & Probability Letters, Elsevier, vol. 3(2), pages 65-69, April.
    2. E. Liebscher, 1990. "Hermite series estimators for probability densities," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 37(1), pages 321-343, December.
    3. Alexandre Leblanc, 2012. "On estimating distribution functions using Bernstein polynomials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(5), pages 919-943, October.
    4. Asma Jmaei & Yousri Slaoui & Wassima Dellagi, 2017. "Recursive distribution estimator defined by stochastic approximation method using Bernstein polynomials," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(4), pages 792-805, October.
    5. Greblicki, W?odzimierz & Pawlak, Miros?aw, 1984. "Hermite series estimates of a probability density and its derivatives," Journal of Multivariate Analysis, Elsevier, vol. 15(2), pages 174-182, October.
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    Cited by:

    1. Stephanou, Michael & Varughese, Melvin, 2021. "Sequential estimation of Spearman rank correlation using Hermite series estimators," Journal of Multivariate Analysis, Elsevier, vol. 186(C).

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