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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.

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.

Suggested Citation

  • Marshall, Jonathan C. & Hazelton, Martin L., 2010. "Boundary kernels for adaptive density estimators on regions with irregular boundaries," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 949-963, April.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:4:p:949-963
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    5. 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.
    6. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 471-480, September.
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    Cited by:

    1. Davies, Tilman M. & Jones, Khair & Hazelton, Martin L., 2016. "Symmetric adaptive smoothing regimens for estimation of the spatial relative risk function," Computational Statistics & Data Analysis, Elsevier, vol. 101(C), pages 12-28.
    2. repec:jss:jstsof:39:i01 is not listed on IDEAS

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