## The Annals of Statistics

### Variable Kernel Density Estimation

#### Abstract

We investigate some of the possibilities for improvement of univariate and multivariate kernel density estimates by varying the window over the domain of estimation, pointwise and globally. Two general approaches are to vary the window width by the point of estimation and by point of the sample observation. The first possibility is shown to be of little efficacy in one variable. In particular, nearest-neighbor estimators in all versions perform poorly in one and two dimensions, but begin to be useful in three or more variables. The second possibility is more promising. We give some general properties and then focus on the popular Abramson estimator. We show that in many practical situations, such as normal data, a nonlocality phenomenon limits the commonly applied version of the Abramson estimator to bias of $O(\lbrack h / \log h\rbrack^2)$ instead of the hoped for $O(h^4)$.

#### Article information

Source
Ann. Statist. Volume 20, Number 3 (1992), 1236-1265.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348768

Digital Object Identifier
doi:10.1214/aos/1176348768

Mathematical Reviews number (MathSciNet)
MR1186249

Zentralblatt MATH identifier
0763.62024

JSTOR