Optimization Problems Connected with Kernel Smoothing

The present paper is dealing with optimization problems arising in the context of kernel estimates of a density and a regression function. Kernel estimates are one of the most popular nonparametric functional estimates. These estimates depend on a bandwidth which controls the smoothness of the estimate and on a kernel which plays a role of a weight function. In this paper we concentrate on a choice of a kernel from different points of view. We deal with the problem of finding smooth optimal kernels minimizing the asymptotic mean squared error and that of smooth minimum variance kernels minimizing the asymptotic variance. The optimization problems under consideration are shown to be equivalent. Then, using the optimal control theory we arrive at the conclusion that smooth optimal kernels have to be polynomials of prescribed degree and they can be expressed in terms of the Gegenbauer polynomials. This result gives the theoretical evidence for using kernels with polynomials shapes. Key-Words: optimization problem, smooth optimal kernel, smooth minimum variance kernel