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Locally optimal window widths for kernel density estimation with large samples

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  • Schucany, William R.

Abstract

The smoothing parameter or window width for a kernel estimator of a probability density at a point has been previously specified to minimize either asymptotic mean square error or asymptotic mean absolute error. In this note the ratio of these two widths is shown to be a constant for all kernels and density functions that satisfy the usual smoothness conditions. The fact that this ratio equals 0.985 supports recent comment that in this context these two error criteria do not yield large-sample results that differ by any meaningful amount. Isolated points at which the dominant term of the conventional bias expansion vanishes are examined. Consideration of additional terms and continuity leads to the conclusion that bias is adequately modeled by a multiple of a single rate in all large but finite sample sizes. In practice, for instance, at inflection points with a second-order kernel the abrupt change in exponent from 1/5 to 1/9 is not necessarily a good representation.

Suggested Citation

  • Schucany, William R., 1989. "Locally optimal window widths for kernel density estimation with large samples," Statistics & Probability Letters, Elsevier, vol. 7(5), pages 401-405, April.
    • Handle: RePEc:eee:stapro:v:7:y:1989:i:5:p:401-405
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    Cited by:

    1. 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.
    2. Farmen, Mark & Marron, J. S., 1999. "An assessment of finite sample performance of adaptive methods in density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 30(2), pages 143-168, April.
    3. Gaoyuan Wang & Yixuan Wang & Yangli Li & Tian Chen, 2023. "Identification of Urban Clusters Based on Multisource Data—An Example of Three Major Urban Agglomerations in China," Land, MDPI, vol. 12(5), pages 1-25, May.
    4. Camerlenghi, F. & Capasso, V. & Villa, E., 2014. "On the estimation of the mean density of random closed sets," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 65-88.

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