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A note on the integrated squared error of a kernel density estimator in non-smooth cases

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  • van Es, Bert

Abstract

Let X1, ... , Xn be a random sample from a distribution on the real line with an unknown density f. We discuss the performance of the kernel density estimator of the density f. The properties of kernel estimators in cases where the density f to be estimated is sufficiently smooth are well known. Instead we focus on estimation problems where f is non-smooth, i.e. f is allowed to have a finite number of jumps or kinks. Thus the robustness properties of the kernel estimator against unfulfilled smoothness assumptions are illustrated. After a review of properties of the mean integrated squared error we present a central limit theorem for the integrated squared error. This theorem extends results of Bickel, Rosenblatt and Hall. Finally, the distance between the bandwidth minimizing the integrated squared error and the bandwidth which minimizes the mean integrated squared error is discussed.

Suggested Citation

  • van Es, Bert, 1997. "A note on the integrated squared error of a kernel density estimator in non-smooth cases," Statistics & Probability Letters, Elsevier, vol. 35(3), pages 241-250, October.
  • Handle: RePEc:eee:stapro:v:35:y:1997:i:3:p:241-250
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    References listed on IDEAS

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    1. Hall, Peter, 1984. "Central limit theorem for integrated square error of multivariate nonparametric density estimators," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 1-16, February.
    2. B. Ghosh & Wei-Min Huang, 1992. "Optimum bandwidths and kernels for estimating certain discontinuous densities," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(3), pages 563-577, September.
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    Cited by:

    1. Cuevas, Antonio & Fraiman, Ricardo, 1998. "On visual distances in density estimation: the Hausdorff choice," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 333-341, November.

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