Derivation of Equivalent Kernel for General Spline Smoothing: A Systematic Approach

We consider first the spline smoothing nonparametric estimation with variable smoothing parameter and arbitrary design density function and show that the corresponding equivalent kernel can be approximated by the Green's function of a certain linear differential operator. Furthermore, we propose to use the standard (in applied mathematics and engineering) method for asymptotic solution of linear differential equations, known as the WKB-method, for systematic derivation of asymptotically equivalent kernel in this general case. The corresponding results for polynomial splines are a special case of the general solution. Then, we show how these ideas can be directly extended to the very general L-spline smoothing. 3 The author responsible for correspondence y Part of the work was done while the author was in the School of Mathematics, University of Bristol, UK 3 Key words and phrases. Green's function, L-smoothing spline, nonparametric regression, variable smoothing parameter, WKB-m...