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Arbitrariness of the pilot estimator in adaptive kernel methods

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  • Abramson, Ian S.

Abstract

Consider estimating a smooth p-variate density f at 0 using the classical kernel estimator fn(0) = n-1 [Sigma]ibn-pw(bn-1Xi) based on a sample {Xi} from f. Under familiar conditions, assigning bn = bn-1/(4 + p) gives the best MSE decay rate O(n-4/(4 + p), but the optimal b, b* say, depends on f through its second derivatives, raising a feasibility objection to its use. By prescribing a pilot estimate of b* based on the same sample, Woodroofe has shown that there need be asymptotically no loss as against knowing the constant exactly, but his proposal is critically dependent on achieving a certain consistency rate for b*. Admitting a minor change in the risk function, we show by a tightness argument applied to the error process that any consistent estimator of b* may be used to achieve the same performance.

Suggested Citation

  • Abramson, Ian S., 1982. "Arbitrariness of the pilot estimator in adaptive kernel methods," Journal of Multivariate Analysis, Elsevier, vol. 12(4), pages 562-567, December.
  • Handle: RePEc:eee:jmvana:v:12:y:1982:i:4:p:562-567
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    Cited by:

    1. Hu, Shuowen & Poskitt, D.S. & Zhang, Xibin, 2012. "Bayesian adaptive bandwidth kernel density estimation of irregular multivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 56(3), pages 732-740.
    2. Zhenyu Jiang & Nengxiang Ling & Zudi Lu & Dag Tj⊘stheim & Qiang Zhang, 2020. "On bandwidth choice for spatial data density estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 817-840, July.
    3. Ziegler Klaus, 2006. "On local bootstrap bandwidth choice in kernel density estimation," Statistics & Risk Modeling, De Gruyter, vol. 24(2), pages 291-301, December.
    4. Sain, Stephan R., 2002. "Multivariate locally adaptive density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 39(2), pages 165-186, April.
    5. Dimitrios Bagkavos, 2008. "Transformations in hazard rate estimation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 721-738.
    6. Cristóbal, J. A. & Alcalá, J. T., 1998. "Error Process Indexed by Bandwidth Matrices in Multivariate Local Linear Smoothing," Journal of Multivariate Analysis, Elsevier, vol. 66(2), pages 207-236, August.
    7. Daniel Ting & Guoli Wang & Maxim Shapovalov & Rajib Mitra & Michael I Jordan & Roland L Dunbrack Jr, 2010. "Neighbor-Dependent Ramachandran Probability Distributions of Amino Acids Developed from a Hierarchical Dirichlet Process Model," PLOS Computational Biology, Public Library of Science, vol. 6(4), pages 1-21, April.
    8. Stefano Magrini, 2007. "Analysing Convergence through the Distribution Dynamics Approach: Why and how?," Working Papers 2007_13, Department of Economics, University of Venice "Ca' Foscari".
    9. Lin, Yan-Hui & Jiao, Xin-Lei, 2021. "Adaptive Kernel Auxiliary Particle Filter Method for Degradation State Estimation," Reliability Engineering and System Safety, Elsevier, vol. 211(C).

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