## The Annals of Statistics

### On Global Properties of Variable Bandwidth Density Estimators

Peter Hall

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348655

Digital Object Identifier
doi:10.1214/aos/1176348655

Mathematical Reviews number (MathSciNet)
MR1165591

Zentralblatt MATH identifier
0785.62040

JSTOR