Spline estimation and variable selection for single-index prediction models with diverging number of index parameters

• We focus on model selection for high-dimensional single-index prediction models.
• We propose a penalized variable selection method with SCAD penalty.
• Our method possesses an oracle property when the number of predictors diverges.
• Our method is robust against model misspecifications.
• Our method can be applied for both independent and time series data.

Single-index models are useful and fundamental tools for handling “curse of dimensionality” problems in nonparametric regression. Along with that, variable selection also plays an important role in such model building process when the index vectors are high-dimensional. Several procedures have been developed for estimation and variable selection for single-index models when the number of index parameters is fixed. In many high-dimensional model selection problems, the number of parameters is increasing along with the sample size. In this work, we consider weakly dependent data and propose a class of variable selection procedures for single-index prediction models, which are robust against model misspecifications. We apply polynomial spline basis function expansion and smoothly clipped absolute deviation penalty to perform estimation and variable selection in the framework of a diverging number of index parameters. Under stationary and strong mixing conditions, the proposed variable selection method is shown to have the “oracle” property when the number of index parameters tends to infinity as the sample size increases. A fast and efficient iterative algorithm is developed to estimate parameters and select significant variables simultaneously. The finite sample behavior of the proposed method is evaluated with simulation studies and illustrated by the river flow data of Iceland.