Journal of the American Statistical Association Linear or Nonlinear? Automatic Structure Discovery for Partially Linear Models Linear or Nonlinear? Automatic Structure Discovery for Partially Linear…

By H. Hao, Zhang, Guang Cheng, Yufeng Liu, Guanggen Zhang, Cheng, Yufeng Cheng, Liu

175 citations recorded by the source corpus.

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.

Subfields: Analytical Chemistry, Statistics and Probability

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Citations in the corpus

Fused Lasso Additive Model (2014, methodology): “Flexible additive modeling in high dimensions has been an active area of research in recent years [Avalos et al., 2007, Huang et al., 2010, Li and Liu, 2014, Lin and Zhang, 2006, Sardy and Tseng, 2004, Wood et al., 2014, Zhang et al., 2011].”
Semiparametric Regression Pursuit (2012, methodology): “Recently, Zhang, Cheng, and Liu (2011) proposed a novel method for de- termining the zero, linear, and nonlinear components in partially linear models.”
Nonparametric independence screening and structure identification for ultra-high dimensional longitudinal data (2013, methodology): “In addition, structure identification was considered for partially linear additive models by [38], and Cox proportional hazard models with varying coefficients by [36, 19].”
GENERALIZED ADDITIVE PARTIAL LINEAR MODELS WITH HIGH-DIMENSIONAL COVARIATES (2013, background): “Recently Zhang, Cheng, and Liu (2011) have carried out an interesting study on automatic discovery of partially linear structure in semiparametric models.”
Identification of Partially Linear Structure in Additive Models with an Application to Gene Expression Prediction from Sequences (2012, methodology): “In Zhang, Cheng, and Liu (2011), the authors used smoothing splines to decompose a nonparametric component into a linear part and another part orthogonal to it.”