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

We address the problem of density estimation with "Lₛ-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding Lₛ-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 Lₛ-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].