The Consistency of Automatic Kernel Density Estimates

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.