Multivariate Local Fitting with General Basis Functions

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.