Partially linear structure identification in generalized additive models with NP-dimensionality

Separation of the linear and nonlinear components in additive models based on penalized likelihood has received attention recently. However, it remains unknown whether consistent separation is possible in generalized additive models, and how high dimensionality is allowed. In this article, we study the doubly SCAD-penalized approach for partial linear structure identification problems of non-polynomial (NP) dimensionality and demonstrate its oracle property. In particular, if the number of nonzero components is fixed, the dimensionality of the total number of components can be of order exp{nd/(2d+1)} where d is the smoothness of the component functions. Under such dimensionality assumptions, we show that with probability approaching one, the proposed procedure can correctly identify the zero, linear, and nonlinear components in the model. We further establish the convergence rate of the estimator for the nonlinear component and the asymptotic normality of the estimator for the linear component. Performance of the proposed method is evaluated by simulation studies. The methods are demonstrated by analyzing a gene data set.