On a Nadaraya-Watson Estimator with Two Bandwidths

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 while choosing very small bandwidths. It performs even better than the ratio of the two best estimators of the numerator and the denominator of the collection, for which larger bandwidth are to be chosen.