American Journal of Theoretical and Applied Statistics

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Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency

Received: 08 January 2016    Accepted: 23 January 2016    Published: 16 February 2016
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Abstract

Nonparametric density estimation, based on kernel-type estimators, is a very popular method in statistical research, especially when we want to model the probabilistic or stochastic structure of a data set. In this paper, we investigate the asymptotic confidence bands for the distribution with kernel-estimators for some types of divergence measures (Rényi-α and Tsallis-α divergence). Our aim is to use the method based on empirical process techniques, in order to derive some asymptotic results. Under different assumptions, we establish a variety of fundamental and theoretical properties, such as the strong consistency of an uniform-in-bandwidth of the divergence estimators. We further apply the previous results in simulated examples, including the kernel-type estimator for Hellinger, Bhattacharyya and Kullback-Leibler divergence, to illustrate this approach, and we show that that the method performs competitively.

DOI 10.11648/j.ajtas.20160501.13
Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 1, January 2016)
Page(s) 13-22
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Divergence Measures, Kernel Estimation, Strong Uniform, Consistency

References
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Author Information
  • Departement de Mathématiques et Informatique, Faculté des Sciences et Technique, Université Cheikh Anta Diop, Dakar, Sénégal

  • Departement de Mathématiques et Informatique, Faculté des Sciences et Technique, Université Cheikh Anta Diop, Dakar, Sénégal

  • Sciences Appliquées et Technologie, Unité de Formation et de Recherche, Université Gaston Berger, Saint-Louis, Sénégal

  • Département de Techniques Quantitatives, Faculté des Sciences Economiques et de Gestion, Université Cheikh Anta Diop , Dakar, Sénégal

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  • APA Style

    Hamza Dhaker, Papa Ngom, El Hadji Deme, Pierre Mendy. (2016). Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency. American Journal of Theoretical and Applied Statistics, 5(1), 13-22. https://doi.org/10.11648/j.ajtas.20160501.13

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    ACS Style

    Hamza Dhaker; Papa Ngom; El Hadji Deme; Pierre Mendy. Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency. Am. J. Theor. Appl. Stat. 2016, 5(1), 13-22. doi: 10.11648/j.ajtas.20160501.13

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    AMA Style

    Hamza Dhaker, Papa Ngom, El Hadji Deme, Pierre Mendy. Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency. Am J Theor Appl Stat. 2016;5(1):13-22. doi: 10.11648/j.ajtas.20160501.13

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  • @article{10.11648/j.ajtas.20160501.13,
      author = {Hamza Dhaker and Papa Ngom and El Hadji Deme and Pierre Mendy},
      title = {Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {5},
      number = {1},
      pages = {13-22},
      doi = {10.11648/j.ajtas.20160501.13},
      url = {https://doi.org/10.11648/j.ajtas.20160501.13},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajtas.20160501.13},
      abstract = {Nonparametric density estimation, based on kernel-type estimators, is a very popular method in statistical research, especially when we want to model the probabilistic or stochastic structure of a data set. In this paper, we investigate the asymptotic confidence bands for the distribution with kernel-estimators for some types of divergence measures (Rényi-α and Tsallis-α divergence). Our aim is to use the method based on empirical process techniques, in order to derive some asymptotic results. Under different assumptions, we establish a variety of fundamental and theoretical properties, such as the strong consistency of an uniform-in-bandwidth of the divergence estimators. We further apply the previous results in simulated examples, including the kernel-type estimator for Hellinger, Bhattacharyya and Kullback-Leibler divergence, to illustrate this approach, and we show that that the method performs competitively.},
     year = {2016}
    }
    

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  • TY  - JOUR
    T1  - Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency
    AU  - Hamza Dhaker
    AU  - Papa Ngom
    AU  - El Hadji Deme
    AU  - Pierre Mendy
    Y1  - 2016/02/16
    PY  - 2016
    N1  - https://doi.org/10.11648/j.ajtas.20160501.13
    DO  - 10.11648/j.ajtas.20160501.13
    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
    SP  - 13
    EP  - 22
    PB  - Science Publishing Group
    SN  - 2326-9006
    UR  - https://doi.org/10.11648/j.ajtas.20160501.13
    AB  - Nonparametric density estimation, based on kernel-type estimators, is a very popular method in statistical research, especially when we want to model the probabilistic or stochastic structure of a data set. In this paper, we investigate the asymptotic confidence bands for the distribution with kernel-estimators for some types of divergence measures (Rényi-α and Tsallis-α divergence). Our aim is to use the method based on empirical process techniques, in order to derive some asymptotic results. Under different assumptions, we establish a variety of fundamental and theoretical properties, such as the strong consistency of an uniform-in-bandwidth of the divergence estimators. We further apply the previous results in simulated examples, including the kernel-type estimator for Hellinger, Bhattacharyya and Kullback-Leibler divergence, to illustrate this approach, and we show that that the method performs competitively.
    VL  - 5
    IS  - 1
    ER  - 

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