Spline estimator for simultaneous variable selection and constant coefficient identification in high-dimensional generalized varying-coefficient models

In this paper, we are concerned with two common and related problems for generalized varying-coefficient models, variable selection and constant coefficient identification. Starting with a specification of generalized varying-coefficient models assuming possible nonlinear interactions between the index variable and all other predictors, we propose a polynomial-spline based procedure that simultaneously eliminates irrelevant predictors and identifies predictors that do not interact with the index variable. Our approach is based on a double-penalization strategy where two penalty functions are used for these two related purposes respectively, in a single functional. In a "large p, small n" setting, we demonstrate the convergence rates of the estimator under suitable regularity assumptions. Based on its previous success on parametric models, we use the extended Bayesian information criterion (eBIC) to automatically choose the regularization parameters. Finally, post-penalization estimator is proposed to further reduce the bias of the resulting estimator. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed procedures and an application to a leukemia dataset is presented.