Wavelets and Nonparametric Function Estimation

The problem of nonparametric function estimation has received a substantial amount of attention in the statistical literature over the last 15 years. To a very large extent, the literature has described kernel-based convolution smoothing solutions to the problems of probability density estimation and nonlinear regression. Among the subcultures within this literature has been a substantial effort at smoothing spline solutions. In the present paper, we discuss a general function analytic formulation of the problem. We show that a basis which spans L2(R) can be used as a tool for constructing computational algorithms for optimal solutions to the generalized nonparametric function estimation problem. In particular wavelets form a doubly indexed set of basis functions for L2(R) and may be used for computing optimal nonparametric function estimates. We discuss the basic theory of wavelets, and discuss connections of wavelets with multiresolution analysis, sub-band coding and conventional spectral analysis. We demonstrate the construction of compactly supported wavelets and illustrate the fractal character of a simple wavelet. We conclude our paper with a discussion of the relationship of wavelets to nonparametric function estimation, to time series analysis, to signal processing and to fractal geometry.