Introduction to Regression Procedures

Nonparametric Regression

Parametric regression models express the mean of an observation as a function of the regressor variables x 1 comma ellipsis comma x Subscript k Baseline and the parameters beta 1 comma ellipsis comma beta Subscript p Baseline:

normal upper E left-bracket upper Y right-bracket equals f left-parenthesis x 1 comma ellipsis comma x Subscript k Baseline semicolon beta 1 comma ellipsis comma beta Subscript p Baseline right-parenthesis

Not only do nonparametric regression techniques relax the assumption of linearity in the regression parameters, but they also do not require that you specify a precise functional form for the relationship between response and regressor variables. Consider a regression problem in which the relationship between response Y and regressor X is to be modeled. It is assumed that normal upper E left-bracket upper Y Subscript i Baseline right-bracket equals g left-parenthesis x Subscript i Baseline right-parenthesis plus epsilon Subscript i, where g left-parenthesis dot right-parenthesis is an unspecified regression function. Two primary approaches in nonparametric regression modeling are as follows:

  • Approximate g left-parenthesis x Subscript i Baseline right-parenthesis locally by a parametric function that is constructed from information in a local neighborhood of x Subscript i.

  • Approximate the unknown function g left-parenthesis x Subscript i Baseline right-parenthesis by a smooth, flexible function and determine the necessary smoothness and continuity properties from the data.

The SAS/STAT procedures ADAPTIVEREG, GAM, GAMPL, LOESS, and TPSPLINE fit nonparametric regression models by one of these methods.

Last updated: December 09, 2022