  Recent contributions to kernel smoothing show that the performance of
cross-validated bandwidth selectors improve significantly from indirectness.
Indirect crossvalidation first estimates the classical cross-validated
bandwidth from a more rough and difficult smoothing problem than the original
one and then rescales this indirect bandwidth to become a bandwidth of the
original problem. The motivation for this approach comes from the observation
that classical crossvalidation tends to work better when the smoothing problem
is difficult. In this paper we find that the performance of indirect
crossvalidation improves theoretically and practically when the polynomial
order of the indirect kernel increases, with the Gaussian kernel as limiting
kernel when the polynomial order goes to infinity. These theoretical and
practical results support the often proposed choice of the Gaussian kernel as
indirect kernel. However, for do-validation our study shows a discrepancy
between asymptotic theory and practical performance. As for indirect
crossvalidation, in asymptotic theory the performance of indirect do-validation
improves with increasing polynomial order of the used indirect kernel. But this
theoretical improvements do not carry over to practice and the original
do-validation still seems to be our preferred bandwidth selector. We also
consider plug-in estimation and combinations of plug-in bandwidths and
crossvalidated bandwidths. These latter bandwidths do not outperform the
original do-validation estimator either.
