Smoothing or nonparametric function estimation was one of the largest areas of statistical research in the 1980s, and is now a well-recognized tool for exploratory data analysis. In regression problems, instead of ﬁtting a simple linear model
E(y|x) = β0 + β1x we ﬁt a ‘nonparametric smooth’ or simply a ‘smooth’ to the data
E(y|x) = f(x) where f(x) is an arbitrary smooth function. Smoothness of the function is a key requirement, as otherwise the estimate may have so much variation that it masks interesting underlying patterns. The model is ‘nonparametric’ in that there are no easily interpretable parameters as in a linear model, but as we shall see, the estimation of f(x) implicitly involves some estimation of parameters. One crucial issue in all smoothing problems is how much to smooth; it is a problem that has given rise to many theoretical developments.