ON THE BOUNDARY KERNEL-METHOD FOR NONPARAMETRIC CURVE ESTIMATION NEAREND-POINTS

Kernel estimators for non-parametric function estimation are affected by boundary effects when estimating near an endpoint of the support of the function. A general construction for boundary kernels is presente d, which allows to remove these edge effects. It is shown that common kernel functions which satisfy some mild requirements can be derived a s the solution of a variational problem involving a certain weight fun ction. For the solutions of this variational problem, an explicit repr esentation in polynomials which are orthogonal with respect to this '' asssociated weight function'' is found; thus any common kernel functio n can be represented as a product of ''an associated weight function'' and an orthogonal expansion. It is demonstrated how this variational problem and its solution can be extended to cover boundary kernels. Th e resulting explicit construction of boundary kernels includes kernels with compact as well as non-compact support, and examples are present ed demonstrating the corresponding boundary kernels for compactly supp orted polynomial kernels, Gaussian kernels with unbounded support, and related analytical kernel functions.