Nonparametric Regression Methods

Besides kernel estimators, commonly used nonparametric regression estimators are local least squares estimators and smoothing splines. Besides these estimators, we also discuss orthogonal series estimators which have been applied mainly in density estimation. All these estimators are localized weighted averages of the data, i.e. linear in the observations (Y_i). The general form is $$ \hat g{\text{L}}\left( {\text{t}} \right) = \sum\limits_{i = 1}^n {W_i } ,{\text{n}}\left( {\text{t}} \right){\text{Y}}_i ,{\text{n}} $$ with weight functions W_i,n(t), and different estimates differ only with respect to the weight functions. As we will see, the estimators considered do not differ too much and asymptotically they are all equivalent to more or less complicated kernel estimators. Therefore, kernel estimators are very general and also the method which is most easily understood intuitively.