Regularized estimation for the least absolute relative error models with a diverging number of covariates

This paper considers the variable selection for the least absolute relative error (LARE) model, where the dimension of model, pnpn, is allowed to increase with the sample size nn. Under some mild regular conditions, we establish the oracle properties, including the consistency of model selection and the asymptotic normality for the estimator of non-zero parameter. An adaptive weighting scheme is considered in the regularization, which admits the adaptive Lasso, SCAD and MCP penalties by linear approximation. The theoretical results allow the dimension diverging at the rate pn=o(n1/2)pn=o(n1/2) for the consistency and pn=o(n1/3)pn=o(n1/3) for the asymptotic normality. Furthermore, a practical variable selection procedure based on least squares approximation (LSA) is studied and its oracle property is also provided. Numerical studies are carried out to evaluate the performance of the proposed approaches.