Linear or Nonlinear? Automatic Structure Discovery for Partially Linear Models.

TitleLinear or Nonlinear? Automatic Structure Discovery for Partially Linear Models.
Publication TypeJournal Article
Year of Publication2011
AuthorsZhang, Hao Helen, Guang Cheng, and Yufeng Liu
JournalJ Am Stat Assoc
Date Published2011 Sep 01

Partially linear models provide a useful class of tools for modeling complex data by naturally incorporating a combination of linear and nonlinear effects within one framework. One key question in partially linear models is the choice of model structure, that is, how to decide which covariates are linear and which are nonlinear. This is a fundamental, yet largely unsolved problem for partially linear models. In practice, one often assumes that the model structure is given or known and then makes estimation and inference based on that structure. Alternatively, there are two methods in common use for tackling the problem: hypotheses testing and visual screening based on the marginal fits. Both methods are quite useful in practice but have their drawbacks. First, it is difficult to construct a powerful procedure for testing multiple hypotheses of linear against nonlinear fits. Second, the screening procedure based on the scatterplots of individual covariate fits may provide an educated guess on the regression function form, but the procedure is ad hoc and lacks theoretical justifications. In this article, we propose a new approach to structure selection for partially linear models, called the LAND (Linear And Nonlinear Discoverer). The procedure is developed in an elegant mathematical framework and possesses desired theoretical and computational properties. Under certain regularity conditions, we show that the LAND estimator is able to identify the underlying true model structure correctly and at the same time estimate the multivariate regression function consistently. The convergence rate of the new estimator is established as well. We further propose an iterative algorithm to implement the procedure and illustrate its performance by simulated and real examples. Supplementary materials for this article are available online.

Alternate JournalJ Am Stat Assoc
Original PublicationLinear or nonlinear? Automatic structure discovery for partially linear models.
PubMed ID22121305
PubMed Central IDPMC3222957
Grant ListP01 CA142538 / CA / NCI NIH HHS / United States
R01 CA085848 / CA / NCI NIH HHS / United States
R01 CA149569 / CA / NCI NIH HHS / United States
R01 CA149569-02 / CA / NCI NIH HHS / United States