Component Identification and Estimation in Partial Linear Regression by Structural Adaptation

• Activity Number: 54
• Type: Contributed
• Date/Time: Sunday, August 3, 2003 : 4:00 PM to 5:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Alexander M. Samarov*+ and Vladimir Spokoiny and Celine Vial
• Companies: Massachusetts Institute of Technology and Humboldt University and ENSAI
• Keywords: partial linear regression ; structural adaptation ; selection of linear variables

Abstract

We propose a new method of analysis of partial linear regression models in which the nonlinear component is possibly multivariate and completely unspecified. The target of analysis is identification of the set of regressors that enter nonlinearly and the complete estimation of the model, including slope coefficients of the linear component and the link function of the nonlinear component. The procedure also allows for selecting the significant regression variables. As a by-product, we develop a test of linear hypothesis against a partially linear alternative, or more generally, a test that the nonlinear component is M-dimensional for M=0,1,2, . . . . The approach proposed in this work is fully adaptive to the unknown model structure and applies under mild conditions on the model. The only important assumption is that the dimensionality of the nonlinear component is relatively small. The theoretical results indicate that the procedure provides a prescribed level of the identification error and estimates the linear component with the accuracy of the order n^{-1/2}. A numerical study demonstrates good performance of the method even for small or moderate sample sizes.