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2006 Weighted uniform consistency of kernel density estimators with general bandwidth sequences
Julia Dony, Uwe Einmahl
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Electron. J. Probab. 11: 844-859 (2006). DOI: 10.1214/EJP.v11-354


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 Gin, Koltchinskii and Zinn (2004) 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.


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Julia Dony. Uwe Einmahl. "Weighted uniform consistency of kernel density estimators with general bandwidth sequences." Electron. J. Probab. 11 844 - 859, 2006.


Accepted: 24 September 2006; Published: 2006
First available in Project Euclid: 31 May 2016

zbMATH: 1107.62030
MathSciNet: MR2261055
Digital Object Identifier: 10.1214/EJP.v11-354

Primary: 60B12
Secondary: 60F15 , 62G07

Keywords: Convergence rates , empirical process , kernel density estimator , uniform in bandwidth , weighted uniform consistency


Vol.11 • 2006
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