Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

Kernel Estimation: the Equivalent Spline Smoothing Method

Wolfgang K. Härdle
Michael Nussbaum

Abstract:
Among nonparametric smoothers, there is a well-known correspondence between
kernel and Fourier series methods, pivoted by the Fourier transform of the
kernel. This suggests a similar relationship between kernel and spline
estimators. A known special case is the result of Silverman (1984) on the
effective kernel for the classical Reinsch-Schoenberg smoothing spline in the
nonparametric regression model. We present an extension by showing that a large
class of kernel estimators have a spline equivalent, in the sense of identical
asymptotic local behaviour of the weighting coefficients. This general class of
spline smoothers includes also the minimax linear estimator over Sobolev
ellipsoids. The analysis is carried out for piecewise linear splines and
equidistant design.

Keywords:
Kernel estimator, spline smoothing, filtering coefficients, differential
operator, Green's function approximation, asymptotic minimax spline

JEL Classification:
C00