Local Fitting with General Basis Functions

Local polynomial modelling can be seen as a local fit of the data
against the basis functions 1, x, ... , x^p. In this paper we
extend this method to a wide range of other basis functions. We
will focus on the power basis, i.e. a basis which consists of the
powers of an arbitrary function, and derive an extended Taylor
theorem for this basis. We describe the estimation procedure and
calculate asymptotic expressions for bias and variance of this
local basis estimator. We apply this method to a simulated data set
for various basis functions and propose a data-driven method to
find a suitable basis function in each situation.