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arXiv · 1006.0971

On the estimation of smooth densities by strict probability densities at optimal rates in sup-norm

Evarist Giné·Hailin Sang
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Abstract

It is shown that the variable bandwidth density estimator proposed by McKay (1993a and b) following earlier findings by Abramson (1982) approximates density functions in $C^4(\mathbb R^d)$ at the minimax rate in the supremum norm over bounded sets where the preliminary density estimates on which they are based are bounded away from zero. A somewhat more complicated estimator proposed by Jones McKay and Hu (1994) to approximate densities in $C^6(\mathbb R)$ is also shown to attain minimax rates in sup norm over the same kind of sets. These estimators are strict probability densities.

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Evarist Giné, Hailin Sang. 2010-06-04. On the estimation of smooth densities by strict probability densities at optimal rates in sup-norm. https://doi.org/10.1214/12-imscoll910

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