Weighted uniform consistency of kernel density estimators with general bandwidth sequences

Julia Dony (Free University of Brussels (VUB))
Uwe Einmahl (Free University of Brussels (VUB))

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.

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Pages: 844-859

Publication Date: September 24, 2006

DOI: 10.1214/EJP.v11-354

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