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

2011; Institute of Mathematical Statistics; Volume: 39; Issue: 3 Linguagem: Inglês

10.1214/11-aos883

2168-8966

Alexander Goldenshluger, Oleg Lepski,

Control Systems and Identification

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].