Component identification and estimation in nonlinear high-dimensional regression models by structural adaptation

This article proposes a new method of analysis of a partially linear model whose nonlinear component is completely unknown. The target of analysis is identification of the set of regressors that enter in a nonlinear way in the model function, and complete estimation of the model, including slope coefficients of the linear component and the link function of the nonlinear component The procedure also allows selection of the significant regression variables. We also 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 article 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 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 accuracy of order n ^-1/2. A numerical study demonstrates a very good performance of the method for even small or moderate sample sizes.