Some results in nonparametric regression

For nonparametric regression, the two most popular methods for constructing kernel estimators, involving choosing weights by kernel evaluation or by subinterval integration, are compared for their asymptotic mean square errors. Their performance is quite different when the design points are serious departures from equal spacing, or when the design points are randomly chosen.;For choosing the bandwidth in nonparametric regression, the ordinary cross-validation provides poor bandwidth estimates when the observations are correlated. In the case of the short range dependent observations, the modified cross-validation is proposed. Based on the reduction of the asymptotic bias of the bandwidth estimates, the modified cross-validation would provide asymptotically optimal bandwidths. However, the modified cross-validated bandwidth suffers from a large amount of sample variability.;The partitioned cross-validation is applied to reduce the sample variability. However, the partitioned cross-validation provides asymptotically biased bandwidths. The two criteria are compared for their asymptotic mean square errors of the bandwidth estimates. In the simulation study, it is shown that the performance of the two criteria depends on the amount of sample variability.