An Application of the Generalized Cross Validation to the Fundamental Cardinal Spline Estimates

Recently, it showed that the kernel type estimates based on the fundamental cardinal splines were asymptotically optimal for certain functional classes. In order to show it, the asymptotically optimal bandwidth which depend on the parameters of given functional class was introduced. However, in practice, the parameters are hardly ever exactly known. So, a practical and appropriate bandwidth should be driven from the data. In this paper we consider the generalized cross validation(GCV) criterion as a method for selecting a bandwidth. We provide the simple formula of GCV criterion in fundamental cardinal spline estimates. In addition, a numerical example is given for illustrating the usefulness of GCV criterion in fundamental cardinal spline estimates.