On Global Properties of Variable Bandwidth Density Estimators

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.