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Full bandwidth matrix selectors for gradient kernel density estimate


  • Horová, Ivana
  • Koláček, Jan
  • Vopatová, Kamila


The most important factor in multivariate kernel density estimation is a choice of a bandwidth matrix. This choice is particularly important, because of its role in controlling both the amount and the direction of multivariate smoothing. Considerable attention has been paid to constrained parameterization of the bandwidth matrix such as a diagonal matrix or a pre-transformation of the data. A general multivariate kernel density derivative estimator has been investigated. Data-driven selectors of full bandwidth matrices for a density and its gradient are considered. The proposed method is based on an optimally balanced relation between the integrated variance and the integrated squared bias. The analysis of statistical properties shows the rationale of the proposed method. In order to compare this method with cross-validation and plug-in methods the relative rate of convergence is determined. The utility of the method is illustrated through a simulation study and real data applications.

Suggested Citation

  • Horová, Ivana & Koláček, Jan & Vopatová, Kamila, 2013. "Full bandwidth matrix selectors for gradient kernel density estimate," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 364-376.
  • Handle: RePEc:eee:csdana:v:57:y:2013:i:1:p:364-376 DOI: 10.1016/j.csda.2012.07.006

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    References listed on IDEAS

    1. Duong, Tarn & Hazelton, Martin L., 2005. "Convergence rates for unconstrained bandwidth matrix selectors in multivariate kernel density estimation," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 417-433, April.
    2. J. Chacón & T. Duong, 2010. "Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 375-398, August.
    3. Ivana Horová & Jiří Zelinka, 2007. "Contribution to the bandwidth choice for kernel density estimates," Computational Statistics, Springer, vol. 22(1), pages 31-47, April.
    4. Tarn Duong & Martin L. Hazelton, 2005. "Cross-validation Bandwidth Matrices for Multivariate Kernel Density Estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 485-506.
    5. Duong, Tarn & Cowling, Arianna & Koch, Inge & Wand, M.P., 2008. "Feature significance for multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 52(9), pages 4225-4242, May.
    6. Cao, Ricardo & Cuevas, Antonio & Gonzalez Manteiga, Wensceslao, 1994. "A comparative study of several smoothing methods in density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 17(2), pages 153-176, February.
    7. Horová Ivana & Vieu Philippe & Zelinka Jiří, 2002. "Optimal Choice Of Nonparametric Estimates Of A Density And Of Its Derivatives," Statistics & Risk Modeling, De Gruyter, vol. 20(1-4), pages 355-378, April.
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    Cited by:

    1. Zougab, Nabil & Adjabi, Smail & Kokonendji, Célestin C., 2014. "Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 28-38.


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