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

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