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Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality. (English) Zbl 1234.62035
Summary: 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 by Goldenshluger and Lepski [Ann. Probab. 39, No. 6, 2318–2384 (2011; Zbl 1238.60023)].

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
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