Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality

Abstract

We address the problem of density estimation with ${\mathbb{L}}_s$-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding ${\mathbb{L}}_s$-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol'skii classes. The main technical tools used in our derivations are uniform bounds on the ${\mathbb{L}}_s$-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].

Published in at http://dx.doi.org/10.1214/11-AOS883 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Universal pointwise selection rule in multivariate function estimation
• Uniform bounds for norms of sums of independent random functions

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