Boundary Effects in Nonparametric Curve Estimation Models

Abstract

Boundary effects disturb nonparametric curve estimates, especially of derivatives, near the boundaries of the support of the curve. Modifications of estimates near the boundaries are necessary in order to obtain global asymptotic results as well as a satisfying finite sample behavior. We describe such modifications exhibiting different degrees of smoothness and derive the rate of a.s. convergence of the supremal deviation of nonparametric kernel regression, supremum taken over the deviation on the whole interval of support of the function to be estimated. A “reference kernel method” is proposed which allows the construction of modified kernels of different degrees of smoothness. This method requires substantially less computations than previous approaches.