Nonparametric Regression Analysis

Regression analysis, broadly construed, traces the conditional distribution -- or some aspect of the conditional distribution, such as its mean -- of a dependent variable (Y) as a function of one or more independent variables (X's). Regression analysis as it is usually applied is much more restrictive -- assuming that the mean of Y is a linear function of the X's, that the conditional distribution of Y given the X's is a normal distribution, and that the conditional variance of Y is constant. These assumptions lead naturally to linear least-squares estimation.

Nonparametric regression makes minimal assumptions about the dependence of the average Y on the X's. This short course will introduce nonparametric regression estimators both for simple- regression analysis (a single X) -- also called scatterplot smoothers -- and for multiple-regression analysis (several X's). I will describe naive binning estimators, kernel (local weighted averaging) estimators, local-polynomial ("lowess") estimators, and additive nonparametric regression models. There will be some consideration of methods of statistical inference for nonparametric regression, analogous to the methods employed for linear least squares.

A background in linear least-squares regression will be assumed. Most of the material will be presented at an elementary mathematical and statistical level. Some of the material on statistical inference requires knowledge of matrix algebra. I will limit this more demanding material to the last lecture.