## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 12, Number 4 (1984), 1231-1249.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346789

Digital Object Identifier
doi:10.1214/aos/1176346789

Mathematical Reviews number (MathSciNet)
MR760685

Zentralblatt MATH identifier
0569.62032

JSTOR