Component identification and estimation in nonlinear high-dimensional regression models by structural adaption
- Samarov, Alexander
- Spokoiny, Vladimir
- Vial, Celine
2010 Mathematics Subject Classification
- 62H30 62J02
- structural adaptation, partially linear model, component analysis
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 which enter in a nonlinear way in the model function, 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 $, M=0,1,2,ldots ,$. par 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 the accuracy of order $, n^-1/2 ,$. A numerical study demonstrates a very good performance of the method even for small or moderate sample sizes.
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