Multivariate Local Fitting with General Basis Functions

Abstract

In this paper we combine the concepts of local smoothing and fitting with basis functions for multivariate predictor variables. We start with arbitrary basis functions and show that the asymptotic variance at interior points is independent of the choice of the basis. Moreover we calculate the asymptotic variance at boundary points. We are not able to compute the asymptotic bias since a Taylor theorem for arbitrary basis functions does not exist. For this reason we focus on basis functions without interactions and derive a Taylor theorem which covers this case. This theorem enables us to calculate the asymptotic bias for interior as well as for boundary points. We demonstrate how advantage can be taken of the idea of local fitting with general basis functions by means of a simulated data set, and also provide a data-driven tool to optimize the basis.