The Annals of Statistics

Variable Kernel Density Estimation

George R. Terrell and David W. Scott

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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

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

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Kernel estimators adaptive estimation nearest-neighbor estimators balloongrams nonparametric smoothing


Terrell, George R.; Scott, David W. Variable Kernel Density Estimation. Ann. Statist. 20 (1992), no. 3, 1236--1265. doi:10.1214/aos/1176348768.

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