A note on the weighted bootstrap approximation of the Bickel-Rosenblatt Statistic

In this article we propose a weighted bootstrap approximation to the distribution
of the supremum (over compact sets) of the classical Bickel-Rosenblatt statistic
|f_n(t) − f(t)|/\sqrt{f(t)} as well as its “Studentized” version |f_n(t) − f(t)|/\sqrt{f_n(t)},
where f_n is the usual kernel density estimator of the true density f. Follow-
ing Horvath et al. (2000), we showed that the proposed weighted bootstrap
method is consistent (in capturing the true limiting distributions derived by Bickel
and Rosenblatt (1973)). Furthermore, simulation results show that the proposed
weighted bootstrap has a much better finite-sample performance than the results
based on asymptotic theory. For comparison purposes, we also consider Efron’s
(1979) original bootstrap.