  Nonparametric density and regression estimators commonly depend on a
bandwidth. The asymptotic properties of these estimators have been widely
studied when bandwidths are nonstochastic. In practice, however, in order to
improve finite sample performance of these estimators, bandwidths are selected
by data driven methods, such as cross-validation or plug-in procedures. As a
result nonparametric estimators are usually constructed using stochastic
bandwidths. In this paper we establish the asymptotic equivalence in
probability of local polynomial regression estimators under stochastic and
nonstochastic bandwidths. Our result extends previous work by Boente and
Fraiman (1995) and Ziegler (2004).
