Minimax Kernels for nonparametric curve estimation

Minimax kernels for nonparametric curve estimation are investigated. They are defined to be the solutions to the kernel variational problem arising from the asymptotic maximum risk of the kernel density or derivative estimation established in this paper. A δ-perturbation method is employed to solve the kernel variational problem. Such a δ-perturbation method can be used in solving other variational problems such as the variational problem of Gasser, Müller and Mammitzsch (1985). We obtain the explicit expressions of the minimax kernels by an algorithm developed in the Appendix and tabulate the asymptotic relative efficiencies among the minimax kernels, optimal kernels and Gaussian-based kernels for further reference. The minimax kernels are shown to possess not only the minimax property, but also have higher asymptotic efficiency in the conventional, non-minimax sense. As a by-product of our study, the asymptotic minimax risks for the kernel density and derivative estimators are also obtained.