Local Walsh-average regression for single index varying coefficient models

Abstract

The local Walsh-average regression has been shown to be a comparable method to the others in nonparametric regression. In this paper, we develop a new estimation procedure based on local Walsh-average regression for single index varying coefficient models. Under some general assumptions, we establish the asymptotic properties of the proposed estimators for both the parametric and nonparametric parts. We further demonstrate that the proposed estimates have great efficiency gains across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error compared with that of local polynomial regression estimates under the least squares loss. Even in the worst case scenarios, the asymptotic relative efficiency owns a lower bound equaling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric link functions respectively, versus the least squares estimators. A simulation study is conducted to assess the finite sample properties of the proposed procedure, and a real data analysis is followed to further illustrate the application of the proposed methodology.