On completely data-driven bandwidth selection for single-index models

The class of single-index models (SIMs) has become an important tool for nonparametric regression analysis. As with any other nonparametric regression models, the selection of bandwidth plays an important role in the inferences of the SIMs. However, most results in the literature either take the bandwidths as externally given, or require unpractical assumptions or very restrictive conditions for data-driven bandwidths. We examine the asymptotic properties of a popular bandwidth selection method based on cross-validation that is completely data-driven, under much weaker conditions than those assumed in the literature. And we show that the same bandwidth that is optimal for estimating the index vector, can be used for nearly optimal error variance estimation through the method of varying cross-validation. A simulation study is presented to demonstrate the finite sample performance of the proposed procedures, based on which we recommend a simple 2-step procedure for bandwidth selection, index vector estimation, as well as error variance estimation.