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Boundary kernels for adaptive density estimators on regions with irregular boundaries

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  • Marshall, Jonathan C.
  • Hazelton, Martin L.
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    Abstract

    In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 101 (2010)
    Issue (Month): 4 (April)
    Pages: 949-963

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    Handle: RePEc:eee:jmvana:v:101:y:2010:i:4:p:949-963

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    Keywords: Adaptive smoothing Boundary bias Edge effects Kernel estimator Variable bandwidth;

    References

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    1. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer, vol. 52(3), pages 471-480, September.
    2. Ezcurra, Roberto, 2007. "Is there cross-country convergence in carbon dioxide emissions?," Energy Policy, Elsevier, vol. 35(2), pages 1363-1372, February.
    3. Hazelton, Martin L. & Marshall, Jonathan C., 2009. "Linear boundary kernels for bivariate density estimation," Statistics & Probability Letters, Elsevier, vol. 79(8), pages 999-1003, April.
    4. Sain, Stephan R., 2002. "Multivariate locally adaptive density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 39(2), pages 165-186, April.
    5. H. G. Müller & U. Stadtmüller, 1999. "Multivariate boundary kernels and a continuous least squares principle," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(2), pages 439-458.
    6. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
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