The consistency of automatic kernel density estimates (Q1060794)

In this article the authors investigate the Parzen-Rosenblatt kernel density estimator on \(R^ d\) 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, the authors show 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 the theorems proved by the authors.