Weighted Hellinger distance as an error criterion for bandwidth selection in kernel estimation

    Authors

    I. A. Ahmad;A. R. Mugdadi

    Comments

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    Abbreviated Journal Title

    J. Nonparametr. Stat.

    Keywords

    Hellinger distance; integrated mean-square error; kernel estimation; distribution function; probability density function; bandwidth; selection; cross-validation; INTEGRATED SQUARED ERROR; Statistics & Probability

    Abstract

    Ever since the pioneering work of Parzen [ Parzen, E., 1962, On estimation of a probability density function and mode. Annales of Mathematics and Statistics, 33, 1065 - 1076.], the mean-square error (MSE) and its integrated form (MISE) have been used as the criteria of error in choosing the window size in kernel density estimation. More recently, however, other criteria have been advocated as competitors to the MISE, such as the mean absolute deviation or the Kullback - Leibler loss. In this note, we define a weighted version of the Hellinger distance and show that it has an asymptotic form, which is one-fourth the asymptotic MISE under a slightly more stringent smoothness conditions on the density f. In addition, the proposed criteria give rise to a new way for data-dependent bandwidth selection, which is more stable in the sense of having smaller MSE than the usual least-squares cross-validation, biased cross-validation or the plug-in methodologies when estimating f. Analogous results for the kernel distribution function estimate are also presented.

    Journal Title

    Journal of Nonparametric Statistics

    Volume

    18

    Issue/Number

    2

    Publication Date

    1-1-2006

    Document Type

    Article

    Language

    English

    First Page

    215

    Last Page

    226

    WOS Identifier

    WOS:000238876000007

    ISSN

    1048-5252

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