Optimal Smoothing in Single-Index Models

1993/03/01 by Wolfgang Karl Härdle, Wolfgang Hardle, Peter Hall +1 · 9 citations
Mathematics · #Statistical Methods and Bayesian Inference #Statistical Methods and Inference

paper · pdf · doi:10.1214/aos/1176349020

openalex publication_date 1993/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Neural Network-based Partial-Linear Single-Index Models for Environmental Mixtures Analysis
• A Flexible Partially Linear Single Index Proportional Hazards Regression Model for Multivariate Survival Data
• A Single Index Approach to Integrated Species Distribution Modeling for Fisheries Abundance Data
• Deep Nonparametric Regression on Approximate Manifolds: Non-Asymptotic Error Bounds with Polynomial Prefactors
• Single index regression models with randomly left-truncated data
• The recovery of ridge functions on the hypercube suffers from the curse\n of dimensionality
• A semiparametric single-index estimator for a class of estimating equation models
• On the rate of convergence of a neural network regression estimate learned by gradient descent
• Discriminative Learning of Similarity and Group Equivariant\n Representations

Related

• Efficient Estimation in Single Index Models through Smoothing splines
• Linear regression estimation in non-linear single index models
• Conditional regression for single-index models
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