Nonparametric data procedures, by smoothing methods, has been well established as a useful data analytic tool. In particular, theoretical and applied research on nonparametric kemel density estimation has had a noticeable influence on related topics, such as nonparametric regression, nonparametric pattern recognition. Particular application of these nonparametric estimations are crucially dependent on the choice of the bandwidth. Hence various data-driven methods for choosing the bandwidth have been proposed and studied. The most widly studied bandwidth selector is least squares cross-validation. And this methods has attracted many statistical analysis for its practical convient use. But this method consume large amounts of computer time. Even with presentedly computational power, it is all too easy to consume inordinate amounts of computer time by using inefficient algorithms for finding estimates. This article concerns an efficient computational algorithm for this methods when the kernel is symmetric and polynomial functions. In Chapter 2, we discuss for the nonparametric kernel density estimations and we suggest an efficient algorithms for this method. In Chapter 3, we discuss for the nonparametric kernel regression estimations and we suggest an efficient algorithms for this method.