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.