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A Cross-Validation Bandwidth Choice for Kernel Density Estimates with Selection Biased Data


  • Wu, Colin O.


This paper studies the risks and bandwidth choices of a kernel estimate of the underlying density when the data are obtained fromsindependent biased samples. The main results of this paper give the asymptotic representation of the integrated squared errors and the mean integrated squared errors of the estimate and establish a cross-validation criterion for bandwidth selection. This kernel density estimate is shown to be asymptotically superior to many other intuitive kernel density estimates. The data-driven cross-validation bandwidth is shown to be asymptotically optimal in the sense of Stone (1984,Ann. Statist.12, 1285-1297). The finite sample properties of the cross-validation bandwidth are investigated through a Monte Carlo simulation.

Suggested Citation

  • Wu, Colin O., 1997. "A Cross-Validation Bandwidth Choice for Kernel Density Estimates with Selection Biased Data," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 38-60, April.
  • Handle: RePEc:eee:jmvana:v:61:y:1997:i:1:p:38-60

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    References listed on IDEAS

    1. Ahmad, Ibrahim A., 1995. "On multivariate kernel estimation for samples from weighted distributions," Statistics & Probability Letters, Elsevier, vol. 22(2), pages 121-129, February.
    2. Morgenthaler, S. & Vardi, Y., 1986. "Choice-based samples : A non-parametric approach," Journal of Econometrics, Elsevier, vol. 32(1), pages 109-125, June.
    3. Colin Wu & Andrew Mao, 1996. "Minimax kernels for density estimation with biased data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(3), pages 451-467, September.
    4. Marron, James Stephen & Härdle, Wolfgang, 1986. "Random approximations to some measures of accuracy in nonparametric curve estimation," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 91-113, October.
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    Cited by:

    1. José Cristóbal & José Alcalá, 2001. "An overview of nonparametric contributions to the problem of functional estimation from biased data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(2), pages 309-332, December.


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