Robust spline-based variable selection in varying coefficient model

The varying coefficient model is widely used as an extension of the linear regression model. Many procedures have been developed for the model estimation, and recently efficient variable selection procedures for the varying coefficient model have been proposed as well. However, those variable selection approaches are mainly built on the least-squares (LS) type method. Although the LS method is a successful and standard choice in the varying coefficient model fitting and variable selection, it may suffer when the errors follow a heavy-tailed distribution or in the presence of outliers. To overcome this issue, we start by developing a novel robust estimator, termed rank-based spline estimator, which combines the ideas of rank inference and polynomial spline. Furthermore, we propose a robust variable selection method, incorporating the smoothly clipped absolute deviation penalty into the rank-based spline loss function. Under mild conditions, we theoretically show that the proposed rank-based spline estimator is highly efficient across a wide spectrum of distributions. Its asymptotic relative efficiency with respect to the LS-based method is closely related to that of the signed-rank Wilcoxon test with respect to the t test. Moreover, the proposed variable selection method can identify the true model consistently, and the resulting estimator can be as efficient as the oracle estimator. Simulation studies show that our procedure has better performance than the LS-based method when the errors deviate from normality.