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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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    Citations

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    Cited by:

    1. Dimitrios Bagkavos, 2008. "Transformations in hazard rate estimation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 721-738.
    2. 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.
    3. 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.
    4. 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.
    5. 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".
    6. 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.
    7. 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).
    8. Ziegler Klaus, 2006. "On local bootstrap bandwidth choice in kernel density estimation," Statistics & Risk Modeling, De Gruyter, vol. 24(2), pages 1-11, December.
    9. Sain, Stephan R., 2002. "Multivariate locally adaptive density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 39(2), pages 165-186, April.

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