The Consistency of Automatic Kernel Density Estimates
The Annals of Statistics · 1984 · Volume 12 · Issue 4
Abstract
We consider the Parzen-Rosenblatt kernel density estimate on $\mathbb{R}^d$ with data-dependent smoothing factor. Sufficient conditions on the asymptotic behavior of the smoothing factor are given under which the estimate is pointwise consistent almost everywhere for all densities $f$ to be estimated. When the smoothing factor is a function only of the sample size $n$, it is shown that these conditions are also necessary, a generalization of results by Deheuvels. The consistency of various automatic kernel density estimates is a simple consequence of these theorems.
Keywords
60F15, 62G05, automatic kernel estimates, consistency, kernel density estimates, Nonparametric density estimation
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