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Weighted uniform consistency of kernel density estimators with general bandwidth sequences. (English) Zbl 1107.62030

Summary: Let \(f_{n,h}\) be a kernel density estimator of a continuous and bounded \(d\)-dimensional density \(f\). Let \(\psi(t)\) be a positive continuous function such that \(\|\psi f^\beta\|_\infty<\infty\) for some \(0<\beta<1/2\). We are interested in the rate of consistency of such estimators with respect to the weighted sup-norm determined by \(\psi\). This problem has been considered by E. Giné, V. Kolchinskii and J. Zinn [Ann. Probab. 32, No. 38, 2570–2605 (2004; Zbl 1052.62034)] for a deterministic bandwidth \(h_n\). We provide “uniform in \(h\)” versions of some of their results, allowing us to determine the corresponding rates of consistency for kernel density estimators where the bandwidth sequences may depend on the data and/or the location.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 1052.62034
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