Smooth-threshold estimating equations for varying coefficient partially nonlinear models based on orthogonality-projection method

In this paper, a two-stage estimation procedure is proposed for varying coefficient partially nonlinear models, in which the estimates of parametric vector and coefficient functions do not affect each other. Specifically, we first employ an orthogonality-projection-based method for the estimation of parametric coefficient by removing out the varying coefficient parts. In the second stage, we approximate each coefficient function via B-spline basis functions and develop a novel variable selection procedure based on smooth-threshold estimating equations. The proposed procedure can automatically eliminate the irrelevant covariates by setting the corresponding coefficient functions as zero, and simultaneously estimate the nonzero regression components. Besides, this approach not only avoids solving a convex optimization problem that is required in previous variable selection procedures, but also is flexible and easy to implement. Under some mild conditions, the asymptotic theories of the generated estimates are established, including model selection consistency, rate of convergence as well as asymptotic distribution. Finally, some numerical simulations are conducted to examine the finite sample performance of the proposed methodologies, and a real data analysis is followed to further illustrate the application of the methods.