  Accurate approximation of the sampling distribution of nonparametric kernel
density estimators is crucial for many statistical inference problems. Since
these estimators have complex asymptotic distributions, bootstrap methods are
often used for this purpose. With i.i.d. observations, a large literature
exists concerning optimal bootstrap methods which achieve the fastest possible
convergence rate of the bootstrap estimator of the sampling distribution of the
kernel density estimator. With dependent data, such an optimality theory is an
important open problem. We establish a general theory of optimality of the
block bootstrap for kernel density estimation under weak dependence assumptions
which are satisfied by many important time series models. We propose a unified
framework for a theoretical study of a rich class of bootstrap methods which
include as special cases subsampling, Kunsch's moving block bootstrap, Hall's
under-smoothing (UNS) as well as approaches incorporating no (NBC) or explicit
bias correction (EBC). Moreover, we consider their accuracy under a broad
spectrum of choices of the bandwidth $h$, which include as an important special
case the MSE-optimal choice, as well as other under-smoothed choices. Under
each choice of $h$, we derive the optimal tuning parameters and compare optimal
performances between the main subclasses (EBC, NBC, UNS) of the bootstrap
methods.
