The Annals of Statistics

On Global Properties of Variable Bandwidth Density Estimators

Peter Hall

Full-text: Open access


It is argued that mean integrated squared error is not a useful measure of the performance of a variable bandwidth density estimator based on Abramson's square root law. The reason is that when the unknown density $f$ has even moderately light tails, properties of those tails drive the formula for optimal bandwidth, to the virtual exclusion of other properties of $f$. We suggest that weighted integrated squared error be employed as the performance criterion, using a weight function with compact support. It is shown that this criterion is driven by pointwise properties of $f$. Furthermore, weighted squared-error cross-validation selects a bandwidth which gives first-order asymptotic optimality of an adaptive, feasible version of Abramson's variable bandwidth estimator.

Article information

Ann. Statist., Volume 20, Number 2 (1992), 762-778.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62H12: Estimation

Cross-validation density estimation integrated squared error square root law variable bandwidth


Hall, Peter. On Global Properties of Variable Bandwidth Density Estimators. Ann. Statist. 20 (1992), no. 2, 762--778. doi:10.1214/aos/1176348655.

Export citation