Model structure selection in single-index-coefficient regression models

Single-index-coefficient regression models (SICRM) have been proposed and used in the literature for avoiding the “curse of dimensionality”. However, there is no efficient model structure determination methodology for the SICRM. This may cause a tendency to use models that are much larger than required. In this paper, we propose a new procedure for model structure determination in the SICRM; that is, the penalized estimating equations (PEE) for variable selection that combines the “delete-one-component” method and the smoothly clipped absolute deviation penalty. The proposed PEE method can simultaneously identify significant variables of the index and estimate the nonzero coefficients of the index parameters. We also further study testing for nonparametric index-coefficient functions. Asymptotic properties for the proposed estimation procedure have been established. Under the appropriate conditions, we demonstrate that the proposed estimators have the oracle properties. Monte Carlo simulation studies are conducted to assess the finite sample performance of the proposed methods. A real example is analyzed as an illustration.