Statistical Properties of Smoothers

Abstract

The smoothers discussed in Chapter 2 provide very useful descriptions of regression data. However, when we use smoothers to formally estimate a regression function, it becomes important to understand their statistical properties. In this chapter we discuss issues such as mean squared error and sampling distribution of an estimator, and using smoothers to obtain confidence intervals for values of the regression function. We will consider two types of smoothers: Gasser-Müller type kernel estimators and tapered Fourier series. We choose to focus on Gasser-Müller rather than NadarayaWatson type kernel smoothers since the latter have a more complicated bias representation. Among Fourier series estimators the emphasis will be on truncated series estimators, since there are certain practical and theoretical advantages to using an estimator with a discrete smoothing parameter.