Orthogonality-projection-based penalized variable selection for high-dimensional partially linear models

The Smoothly Clipped Absolute Deviation Net variable selection process is introduced for high-dimensional partially linear models. To mitigate the influence of nonparametric components on variable selection and estimation of parametric components, B-spline and orthogonality-projection techniques are employed, thereby constructing the Smoothly Clipped Absolute Deviation Net penalized least squares objective function. The proposed variable selection process not only identifies relevant variables but also simultaneously provides estimates for the selected variables. Additionally, we present estimators for the nonparametric components. Under certain regularization conditions, the variable selection procedure for the proposed parametric components has been proven to exhibit the grouping effect and oracle property, ensuring accurate variable selection. Concurrently, the estimation of nonparametric components achieves the optimal convergence rate for nonparametric estimation. To efficiently implement this process, we propose an algorithm based on local linear approximation and utilize K-fold cross-validation to select the penalty parameter. Our simulation studies and analyses of real data comprehensively cover scenarios where the number of variables p is either less than or far greater than the sample size n.