Rank-based shrinkage estimation for identification in semiparametric additive models

In this paper, we propose a novel and robust procedure for model identification in semiparametric additive models based on rank regression and spline approximation. Under some mild conditions, we establish the theoretical properties of the identified nonparametric functions and the linear parameters. Furthermore, we demonstrate that the proposed rank estimate has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error compared with that of least square estimate. Even in the worst case scenarios, the asymptotic relative efficiency of the proposed rank estimate versus least squares estimate, which is show to have an expression closely related to that of the signed-rank Wilcoxon test in comparison with the t-test, has a lower bound equal to 0.864. Finally, an efficient algorithm is presented for computation and the selections of tuning parameters are discussed. Some simulation studies and a real data analysis are conducted to illustrate the finite sample performance of the proposed method.