Density adjusted kernel smoothers for random design nonparametric regression

Abstract

It has been shown in recent years that quotient (Nadaraya-Watson) and convolution (Priestley-Chao or Gasser-Müller)-type kernel estimators both have distinct disadvantages when applied in random design nonparametric regression settings. Improved asymptotic behavior is achieved by the locally weighted least-squares estimator fitting local lines. We investigate the question whether this supreme asymptotic behavior can be achieved by directly modified versions of the Nadaraya-Watson estimator. It is shown that one modified version, the “Density Adjusted Kernel Smoother (DAKS)” which is introduced here, achieves, in fact, the same desirable asymptotic distribution characteristics as the locally weighted least-squares estimator. This yields an alternative “linearly unbiased” kernel estimator, i.e., the asymptotic bias depends only on the local curvature of the regression function at the point where it is to be estimated.