  In this paper, we are interested in the problem of smoothing parameter
selection in nonparametric curve estimation under dependent errors. We focus on
kernel estimation and the case when the errors form a general stationary
sequence of martingale difference random variables where neither linearity
assumption nor "all moments are finite" are required.We compare the behaviors
of the smoothing bandwidths obtained by minimizing either the unknown average
squared error, the theoretical mean average squared error, a Mallows-type
criterion adapted to the dependent case and the family of criteria known as
generalized cross validation (GCV) extensions of the Mallows' criterion. We
prove that these three minimizers and those based on the GCV family are
first-order equivalent in probability. We give also a normal asymptotic
behavior of the gap between the minimizer of the average square error and that
of the Mallows-type criterion. This is extended to the GCV family.Finally, we
apply our theoretical results to a specific case of martingale difference
sequence, namely the Auto-Regressive Conditional Heteroscedastic (ARCH(1))
process.A Monte-carlo simulation study, for this regression model with ARCH(1)
process, is conducted.
