Optimal Smoothing in Single-Index Models

Abstract

Single-index models generalize linear regression. They have applications to a variety of fields, such as discrete choice analysis in econometrics and dose response models in biometrics, where high-dimensional regression models are often employed. Single-index models are similar to the first step of projection pursuit regression, a dimension-reduction method. In both cases the orientation vector can be estimated root-n consistently, even if the unknown univariate function (or nonparametric link function) is assumed to come from a large smoothness class. However, as we show in the present paper, the similarities end there. In particular, the amount of smoothing necessary for root-n consistent orientation estimation is very different in the two cases. We suggest a simple, empirical rule for selecting the bandwidth appropriate to single-index models. This rule is studies in a small simulation study and an application in binary response models.