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 fn,h be a kernel density estimator of a continuous and bounded d-dimensional
density f. Let ψ(t) be a positive continuous function such that ||ψ
fβ||∞<∞ for some 0<β<1/2. We are interested in the rate
of consistency of such estimators with respect to the weighted sup-norm determined
by ψ. This problem has been considered by Gin, Koltchinskii and Zinn (2004)
for a deterministic bandwidth hn. 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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