Single Index Regression Models with Convex Link

• Activity Number: 261
• Type: Contributed
• Date/Time: Monday, August 10, 2015 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Rohit Patra* and Bodhisattva Sen and Arun Kumar
• Companies: Columbia University and Columbia University and Indian Statistical Institute
• Keywords: Approximately least favorable sub-provided models ; interpolation inequality ; penalized least squares ; semiparametric model ; shape restricted function estimation

Abstract

We consider estimation and inference in a single index regression model with an unknown convex link function. In contrast to the standard approach of using kernel methods, we use shape-constrained splines to estimate the convex link function. We develop a method to compute the penalized least squares estimators (PLSEs) of the parametric and the non-parametric components given i.i.d.~data. We prove the consistency and find the rates of convergence of the estimators. We establish $n^{-1/2}$-rate of convergence and the asymptotic efficiency of the parametric component under mild assumptions. Our analysis of the parametric component is novel as: the PLSEs lie on the ``boundary" of the parameter set (due to the imposed shape constraint), there does not exist a least favorable path through the estimator, and the score of the approximately least favorable path does not satisfy the standard conditions. A finite sample simulation corroborates our asymptotic theory and illustrates the superiority of our procedure over kernel methods, without shape restrictions. The identifiability and existence of the PLSEs are also investigated.