Kernel density estimators have become quite popular in recent years, both as theoretical and practical enterprises, for detecting and displaying distributional structure in populations and ongoing research, mainly concerned with the data-driven choice of the bandwidth h, has made this methodology increasingly more practical. Simultaneously with the kernel density estimators nonparametric regression estimation has become a prominent statistical research topic as a useful tool. Like the discussion of the nonparametric kernel density, the applications of kernel regression always require a crucial choice of bandwidth. Hence various methods have been developed for data-based procedure for choosing the optimal bandwidth. In Chapter 2, we consider kernel density estimation, least-squares cross-validation (CV) and biased cross-validation (BCV). These methods are applied to kernel regression estimation. We proposed, in Chapter 3, the biased cross-validation bandwidth selector BCV. Rice (1984) discussed the relationship of various nonparametric kernel regression bandwidth selectors. Later, in simulations done by Hardle et al. (1988) the selectors, discussed by Rice, performed quite differently from each other. We show that the bandwidth chosen by BCV method proposed in this thesis is optimal in the sense of the asymptotical mean average squared error criterion, and has small sample variability. The simulation results verify that when the underlying regression is sufficiently smooth, the proposed bandwidth is closer to optimal bandwidth $h_m$ (or $h^*$). Because of the assumptions of the underlying regression, there is still room for improvement, but the proposed BCV bandwidth has best performance than other selectors.
