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.

翻译：在回归模型中,我们写下Nadaraya-Watson 的回归函数估计符,作为两个内核估测器的商数,并为分子和分母提出带宽选择方法。我们证明数据驱动估测器和由此得出的比率的风险界限。模拟研究证实,这两个估测器的性能都优于通过对带宽进行交叉校验而获得的性能。然而,出乎意料地发现,单带宽跨校准估测器比前两个好估测器在小噪声环境中的比率要好得多。然而,这两种方法在大噪声模型中都有相似的性能。