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Unified estimation of densities on bounded and unbounded domains

Author

Listed:
  • Kairat Mynbaev

    () (Kazakh-British Technical University)

  • Carlos Martins-Filho

    () (University of Colorado
    IFPRI)

Abstract

Kernel density estimation in domains with boundaries is known to suffer from undesirable boundary effects. We show that in the case of smooth densities, a general and elegant approach is to estimate an extension of the density. The resulting estimators in domains with boundaries have biases and variances expressed in terms of density extensions and extension parameters. The result is that they have the same rates at boundary and interior points of the domain. Contrary to the extant literature, our estimators require no kernel modification near the boundary and kernels commonly used for estimation on the real line can be applied. Densities defined on the half-axis and in a unit interval are considered. The results are applied to estimation of densities that are discontinuous or have discontinuous derivatives, where they yield the same rates of convergence as for smooth densities on $${\mathbb {R}}$$ R .

Suggested Citation

  • Kairat Mynbaev & Carlos Martins-Filho, 2019. "Unified estimation of densities on bounded and unbounded domains," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(4), pages 853-887, August.
    • Handle: RePEc:spr:aistmt:v:71:y:2019:i:4:d:10.1007_s10463-018-0663-z
    DOI: 10.1007/s10463-018-0663-z
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    References listed on IDEAS

    as
    1. Kairat Mynbaev & Carlos Martins-Filho, 2010. "Bias reduction in kernel density estimation via Lipschitz condition," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(2), pages 219-235.
    2. Kairat Mynbaev & Carlos Martins-Filho & Aziza Aipenova, 2016. "A Class of Nonparametric Density Derivative Estimators Based on Global Lipschitz Conditions," Advances in Econometrics, in: Gloria GonzÁlez-Rivera & R. Carter Hill & Tae-Hwy Lee (ed.), Essays in Honor of Aman Ullah, volume 36, pages 591-615, Emerald Publishing Ltd.
    3. McCrary, Justin, 2008. "Manipulation of the running variable in the regression discontinuity design: A density test," Journal of Econometrics, Elsevier, vol. 142(2), pages 698-714, February.
    4. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 471-480, September.
    5. Kuangyu Wen & Ximing Wu, 2015. "An Improved Transformation-Based Kernel Estimator of Densities on the Unit Interval," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 773-783, June.
    6. Malec, Peter & Schienle, Melanie, 2014. "Nonparametric kernel density estimation near the boundary," Computational Statistics & Data Analysis, Elsevier, vol. 72(C), pages 57-76.
    7. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
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    More about this item

    Keywords

    Nonparametric density estimation; Hestenes’ extension; Estimation in bounded domains; Estimation of discontinuous densities;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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