Advertisement
Published in:
2021 | OriginalPaper | Chapter
Profile Least Squares Estimators in the Monotone Single Index Model
Authors : Fadoua Balabdaoui, Piet Groeneboom
Published in: Advances in Contemporary Statistics and Econometrics
Publisher: Springer International Publishing
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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.