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Mise of kernel estimates of a density and its derivatives


  • Singh, Radhey S.


For an integer p [greater-or-equal, slanted] 0, Singh has considered a class of kernel estimators [latin small letter f with hook]~(p) of the pth order derivative [latin small letter f with hook](p) of a density [latin small letter f with hook] and showed how specializations of some of the results there improve the corresponding existing results. In this paper these improved estimators are examined on a global measure of quality of an estimator, namely, the mean integrated square error (MISE) behavior. An upper bound, which can not be tightened any further for a wide class of kernels, is obtained for MISE ([latin small letter f with hook]~(p)). The exact asymptotic value for the same is also obtained. Under two alternative conditions, weaker than those assumed for the two results mentioned above, convergence of MISE ([latin small letter f with hook]~(p)) to zero is proved. Specializations of some of the results here improve the corresponding existing results by weakening the conditions, sharpening the rates of convergence or both.

Suggested Citation

  • Singh, Radhey S., 1987. "Mise of kernel estimates of a density and its derivatives," Statistics & Probability Letters, Elsevier, vol. 5(2), pages 153-159, March.
  • Handle: RePEc:eee:stapro:v:5:y:1987:i:2:p:153-159

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    Cited by:

    1. 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.
    2. 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.
    3. Paul Deheuvels & David Mason, 2004. "General Asymptotic Confidence Bands Based on Kernel-type Function Estimators," Statistical Inference for Stochastic Processes, Springer, vol. 7(3), pages 225-277, October.


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