Locally penalized single-index model using B-splines and spherical coordinates

In this study, we focus on the estimation of the regression function in the single-index model based on B-splines using penalization techniques. We adopt a spherical coordinates reparameterization of an index vector to deal with an identification problem of the single-index model. To provide a spatially adaptive method, two types of penalties are applied to the estimation of the index vector and the regression function. A special penalty called the localized penalty is introduced to handle the sparsity of the index vector using the spherical coordinates, and the total variation penalty is considered to deal with the smoothing function. Using a coordinate descent algorithm with a grid search of the two tuning parameters, the entire solution paths of the index coefficients and the regression functions for tuning parameters can be obtained efficiently. The performance of the proposed estimator is studied through both numerical simulations and real data sets. An R software package pbssim is available.