Rates of strong uniform consistency for multivariate kernel density estimators |
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Authors: | Evarist Gin Armelle Guillou |
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Institution: | a Departments of Mathematics and Statistics, University of Connecticut, Storrs, CT 06269, USA;b Université Paris VI, L.S.T.A., tour 45-55 E3, 4 place Jussieu, 75252, Paris cedex 05, France |
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Abstract: | Let fn denote the usual kernel density estimator in several dimensions. It is shown that if {an} is a regular band sequence, K is a bounded square integrable kernel of several variables, satisfying some additional mild conditions ((K1) below), and if the data consist of an i.i.d. sample from a distribution possessing a bounded density f with respect to Lebesgue measure on Rd, then for some absolute constant C that depends only on d. With some additional but still weak conditions, it is proved that the above sequence of normalized suprema converges a.s. to
. Convergence of the moment generating functions is also proved. Neither of these results require f to be strictly positive. These results improve upon, and extend to several dimensions, results by Silverman 13] for univariate densities. |
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Keywords: | Uniform almost sure rates Non-parametric density estimation Kernel density estimators |
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