  We investigate the discrepancy principle for choosing smoothing parameters
for kernel density estimation. The method is based on the distance between the
empirical and estimated distribution functions. We prove some new positive and
negative results on L_1-consistency of kernel estimators with bandwidths chosen
using the discrepancy principle. Consistency crucially depends on a rather weak
H\"older condition on the distribution function. We also unify and extend
previous results on the behaviour of the chosen bandwidth under more strict
smoothness assumptions. Furthermore, we compare the discrepancy principle to
standard methods in a simulation study. Surprisingly, some of the proposals
work reasonably well over a large set of different densities and sample sizes,
and the performance of the methods at least up to n=2500 can be quite different
from their asymptotic behavior.
