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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 3537564, 7 pages
http://dx.doi.org/10.1155/2016/3537564
Research Article

Linearity Identification for General Partial Linear Single-Index Models

Center of Statistics, Southwestern University of Finance and Economics, Chengdu, China

Received 5 April 2016; Accepted 25 August 2016

Academic Editor: Alessio Merola

Copyright © 2016 Shaogao Lv and Luhong Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Partial linear models, a family of popular semiparametric models, provide us with an interpretable and flexible assumption for modelling complex data. One challenging question in partial linear models is the structure identification for the linear components and the nonlinear components, especially for high dimensional data. This paper considers the structure identification problem in the general partial linear single-index models, where the link function is unknown. We propose two penalized methods based on a modern dimension reduction technique. Under certain regularity conditions, we show that the second estimator is able to identify the underlying true model structure correctly. The convergence rate of the new estimator is established as well.