Kernel Smoothers: An Overview of Curve Estimators for the First Graduate Course in Nonparametric Statistics

Abstract

An introduction to nonparametric regression is accomplished with selected real data sets, statistical graphics and simulations from known functions. It is pedagogically effective for many to have some initial intuition about what the techniques are and why they work. Visual displays of small examples along with the plots of several types of smoothers are a good beginning. Some students benefit from a brief historical development of the topic, provided that they are familiar with other methodology, such as linear regression. Ultimately, one must engage the formulas for some of the linear curve estimators. These mathematical expressions for local smoothers are more easily understood after the student has seen a graph and a description of what the procedure is actually doing. In this article there are several such figures. These are mostly scatterplots of a single response against one predictor. Kernel smoothers have series expansions for bias and variance. The leading terms of those expansions yield approximate expressions for asymptotic mean squared error. In turn these provide one criterion for selection of the bandwidth. This choice of a smoothing parameter is done a rich variety of ways in practice. The final sections cover alternative approaches and extensions. The survey is supplemented with citations to some excellent books and articles. These provide the student with an entry into the literature, which is rapidly developing in traditional print media as well as on line.