Profile Least Squares Estimators in the Monotone Single Index Model

Abstract

We consider least squares estimators of the finite regression parameter $\boldsymbol{\alpha }$ in the single index regression model $Y=\psi (\boldsymbol{\alpha }^T\boldsymbol{X})+\varepsilon$, where $\boldsymbol{X}$ is a d-dimensional random vector, ${\mathbb E}(Y|\boldsymbol{X})=\psi (\boldsymbol{\alpha }^T\boldsymbol{X})$, and $\psi$ is a monotone. It has been suggested to estimate $\boldsymbol{\alpha }$ by a profile least squares estimator, minimizing $\sum _{i=1}^n(Y_i-\psi (\boldsymbol{\alpha }^T\boldsymbol{X}_i))^2$ over monotone $\psi$ and $\boldsymbol{\alpha }$ on the boundary $\mathcal {S}_{d-1}$ of the unit ball. Although this suggestion has been around for a long time, it is still unknown whether the estimate is $\sqrt{n}$-convergent. We show that a profile least squares estimator, using the same pointwise least squares estimator for fixed $\boldsymbol{\alpha }$, but using a different global sum of squares, is $\sqrt{n}$-convergent and asymptotically normal. The difference between the corresponding loss functions is studied and also a comparison with other methods is given.