  In a regression model, we write the Nadaraya-Watson estimator of the
regression function as the quotient of two kernel estimators, and propose a
bandwidth selection method for both the numerator and the denominator. We prove
risk bounds for both data driven estimators and for the resulting ratio. The
simulation study confirms that both estimators have good performances, compared
to the ones obtained by cross-validation selection of the bandwidth. However,
unexpectedly, the single-bandwidth cross-validation estimator is found to be
much better than the ratio of the previous two good estimators, in the small
noise context. However, the two methods have similar performances in models
with large noise.
