High-Dimensional Single-Index Models: Link Estimation and Marginal Inference

TL;DR

This paper introduces a new estimation method for high-dimensional single-index models that accurately estimates the link function, leading to more efficient inference and valid confidence intervals.

Contribution

We develop a novel approach for link function estimation in high-dimensional single-index models, enabling improved parameter estimation and hypothesis testing.

Findings

Proposed estimators are asymptotically normal.

Method provides valid confidence intervals and p-values.

Numerical experiments confirm theoretical properties.

Abstract

This study proposes a novel method for estimation and hypothesis testing in high-dimensional single-index models. We address a common scenario where the sample size and the dimension of regression coefficients are large and comparable. Unlike traditional approaches, which often overlook the estimation of the unknown link function, we introduce a new method for link function estimation. Leveraging the information from the estimated link function, we propose more efficient estimators that are better aligned with the underlying model. Furthermore, we rigorously establish the asymptotic normality of each coordinate of the estimator. This provides a valid construction of confidence intervals and $p$-values for any finite collection of coordinates. Numerical experiments validate our theoretical results.