Generalized varying index coefficient models

In this paper, we propose a new semiparametric model called generalized varying index coefficient models (GVICMs). The GVICM is a generalization of the varying index coefficient model (VICM) proposed by Ma and Song (2014), by allowing for non-Gaussian data and nonlinear link functions. The GVICM serves as a good tool for modeling and assessing nonlinear interaction effects between grouped covariates and the response variable. Our main goal is to estimate the unknown parameters and nonparametric functions. Firstly, we develop a profile spline quasi-likelihood estimation procedure to estimate the regression parameters and nonparametric coefficients in which the nonparametric functions are approximated by BB-spline basis functions. Under some mild conditions, we establish asymptotic normalities of parameter estimations as well as the convergence rates of nonparametric estimators. Secondly, we develop a two-step spline backfitted local quasi-likelihood estimation for achieving asymptotic distribution of nonparametric function. Moreover, the oracle property of the nonparametric estimator is also established by utilizing the two-step estimation approach. Simulation study and a set of real data are carried out to investigate the performance of the proposed method.