Generalized jackknifing and higher order kernels

One way of improving the performance, at least in theory, of kernel estimators of curves such as probability densities, regression functions and spectral densities is to use “higher order” kernel functions. In this paper, we investigate how one might obtain higher order kernels from lower.order ones, and put forward a wide variety of existing and novel formulae under the unifying concept of generalized jackknifing (Schucany, Gray and Owen, 1971). We thus greatly expand on the approach of Schucany and Sommers (1977). Spinoffs include links with more “direct” bias correction methods, a simplified understanding of how the “optimal” polynomial kernels of, for example, Gasser, M ller and Mammitzsch (1985) relate to one another, connections with the Gaussian-based kernels of Wand and Schucany (1990), and many extensions of Terrell and Scott's (1980) method of enforcing nonnegativity in estimates based on higher order kernel ideas