OPTIMAL SMOOTHING IN SINGLE-INDEX MODELS

Single-index models generalize linear regression. They have applicatio ns to a variety of fields, such as discrete choice analysis in econome trics and dose response models in biometrics, where high-dimensional r egression models are often employed. Single-index models are similar t o the first step of projection pursuit regression, a dimension-reducti on method. In both cases the orientation vector can be estimated root- n consistently, even if the unknown univariate function (or nonparamet ric link function) is assumed to come from a large smoothness class. H owever, as we show in the present paper, the similarities end there. I n 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 si ngle-index models. This rule is studied in a small simulation study an d an application in binary response models.