The Annals of Statistics

Optimal Smoothing in Single-Index Models

Wolfgang Hardle, Peter Hall, and Hidehiko Ichimura

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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.

Article information

Ann. Statist., Volume 21, Number 1 (1993), 157-178.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H99: None of the above, but in this section
Secondary: 62H05: Characterization and structure theory

Bandwidth heteroscedastic kernel estimator projection pursuit regression single index model smoothing


Hardle, Wolfgang; Hall, Peter; Ichimura, Hidehiko. Optimal Smoothing in Single-Index Models. Ann. Statist. 21 (1993), no. 1, 157--178. doi:10.1214/aos/1176349020.

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