Electronic Journal of Probability

Weighted uniform consistency of kernel density estimators with general bandwidth sequences

Julia Dony and Uwe Einmahl

Full-text: Open access

Abstract

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.

Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 33, 844-859.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730568

Digital Object Identifier
doi:10.1214/EJP.v11-354

Mathematical Reviews number (MathSciNet)
MR2261055

Zentralblatt MATH identifier
1107.62030

Subjects
Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)
Secondary: 60F15: Strong theorems 62G07: Density estimation

Keywords
kernel density estimator weighted uniform consistency convergence rates uniform in bandwidth empirical process

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Dony, Julia; Einmahl, Uwe. Weighted uniform consistency of kernel density estimators with general bandwidth sequences. Electron. J. Probab. 11 (2006), paper no. 33, 844--859. doi:10.1214/EJP.v11-354. https://projecteuclid.org/euclid.ejp/1464730568


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