On study of kernel regression function polygons

In the case of the random design nonparametric regression, the regression function estimate is produced practically by joining every two consecutive kernel estimates of regression function values by a straight line segment. Hence, it is of polygon type, and is called the kernel regression function polygon (KRFP) in this paper. The KRFP is analyzed by its asymptotic integrated mean square error (AIMSE). This AIMSE precisely quantifies both effects of the kernel function and of the distance between the points on which kernel estimates of regression function values are calculated on the KRFP. By studying the AIMSE, we have the following findings. First of all, if the distance is of smaller order in magnitude than the bandwidth used by the kernel regression function estimator, then Epanechnikov kernel is still the optimal kernel function for the KRFP. Secondly, if the distance is of the same order in magnitude as the bandwidth, then Epanechnikov kernel is no longer optimal for the KRFP. In this case, using th...