Smooth optimum kernel estimators near endpoints

Kernel estimators for smooth curves like density, spectral density or regression functions require modifications when estimating near endpoints of the support, both for practical and asymptotic reasons. The construction of such boundary kernels as solutions of a variational problem is addressed and representations in orthogonal polynomials are given, including explicit solutions for the most important cases. Based on explicit formulae for certain functional of the kernels, it is shown that local bandwidth variation might be indicated near boundaries. Various bandwidth variation schemes are discussed and investigated in a Monte Carlo study.