Simultaneous semiparametric inference for single-index models

TL;DR

This paper develops a comprehensive inference framework for single-index models, establishing asymptotic independence of estimators, optimal convergence rates, and practical tools like confidence bands and hypothesis tests.

Contribution

It introduces a joint inference method for all model parameters in single-index models, including asymptotic independence results and bootstrap procedures.

Findings

Estimator of the link function is asymptotically independent of index-coefficients.

Smoothing spline estimator achieves minimax optimal $L^2$-risk rate.

Proposes simultaneous confidence bands and hypothesis tests for model parameters.

Abstract

In the common partially linear single-index model we establish a Bahadur representation for a smoothing spline estimator of all model parameters and use this result to prove the joint weak convergence of the estimator of the index link function at a given point, together with the estimators of the parametric regression coefficients. We obtain the surprising result that, despite of the nature of single-index models where the link function is evaluated at a linear combination of the index-coefficients, the estimator of the link function and the estimator of the index-coefficients are asymptotically independent. Our approach leverages a delicate analysis based on reproducing kernel Hilbert space and empirical process theory. We show that the smoothing spline estimator achieves the minimax optimal rate with respect to the $L^2$-risk and consider several statistical applications where joint inference on all model parameters is of interest. In particular, we develop a simultaneous confidence band for the link function and propose inference tools to investigate if the maximum absolute deviation between the (unknown) link function and a given function exceeds a given threshold. We also construct tests for joint hypotheses regarding model parameters which involve both the nonparametric and parametric components and propose novel multiplier bootstrap procedures to avoid the estimation of unknown asymptotic quantities.