Key Takeaway: Using the resample version of an empirical bandwidth in nonparametric density estimators does not generally improve accuracy, with no general first-order theoretical improvement expected.

Abstract

We examine the way in which empirical bandwidth choice affects distributional properties of nonparametric density estimators. Two bandwidth selection methods are considered in detail: local and global plug-in rules. Particular attention is focussed on whether the accuracy of distributional bootstrap approximations is appreciably influenced by using the resample version h * , rather than the sample version h, of an empirical bandwidth. It is shown theoretically that, in marked contrast to similar problems in more familiar settings, no general first-order theoretical improvement can be expected when using the resampling version. In the case of local plug-in rules, the inability of the bootstrap to accurately reflect biases of the components used to construct the bandwidth selector means that the bootstrap distribution of h * is unable to capture some of the main properties of the distribution of h. If the second derivative component is slightly undersmoothed then some improvements are possible through using h * , but they would be difficult to achieve in practice. On the other hand, for global plug-in methods, both h and h * are such good approximations to an optimal, deterministic bandwidth that the variations of either can be largely ignored, at least at a first-order level. Thus, for quite different reasons in the two cases, the computational burden of varying an empirical bandwidth across resamples is difficult to justify.