Score estimation in the monotone single index model

TL;DR

This paper demonstrates that solving a specific score equation yields parametric-rate convergence for the monotone single index model's estimator, and introduces a practical, reparametrization-free solution method suitable for high-dimensional data.

Contribution

It shows that the profile least squares estimator converges at the parametric rate when solving a derived score equation, and provides a reparametrization-free approach for high-dimensional settings.

Findings

Score equation solution achieves parametric convergence rate.

Reparametrization-free method simplifies high-dimensional estimation.

Comparison with existing methods highlights advantages in certain scenarios.

Abstract

We consider estimation in the single index model where the link function is monotone. For this model a profile least squares estimator has been proposed to estimate the unknown link function and index. Although it is natural to propose this procedure, it is still unknown whether it produces index estimates which converge at the parametric rate. We show that this holds if we solve a score equation corresponding to this least squares problem. Using a Lagrangian formulation, we show how one can solve this score equation without any reparametrization. This makes it easy to solve the score equations in high dimensions. We also compare our method with the Effective Dimension Reduction (EDR) and the Penalized Least Squares Estimator (PLSE) methods, both available on CRAN as R packages, and compare with link-free methods, where the covariates are ellipticallly symmetric.