Expectile regression for analyzing heteroscedasticity in high dimension

High-dimensional data often display heteroscedasticity and this feature has attracted a lot of attention and discussion. In this paper, we propose regularized expectile regression with SCAD penalty for analyzing heteroscedasticity in high dimension when the error has finite moments. Since the corresponding optimization problem is nonconvex due to the SCAD penalty, we adopt the CCCP (coupling of the concave and convex procedure) algorithm to solve this problem. Under some regular conditions, we can prove that with probability tending to one, the proposed algorithm converges to the oracle estimator after several iterations. We should address that the higher order moment the error has, the higher dimension cardinality our procedure can handle with. If the error follows gaussian or sub-gaussian distribution, our method can be extended to deal with ultra high-dimensional data. Furthermore, by taking different expectile weight level α, we are able to detect heteroscedasticity and explore the entire conditional distribution of the response variable given all the covariates. We investigate the performances of our proposed method through Monte Carlo simulation study and real application and the numerical results show that the resulting estimator by our algorithm enjoys good performance and demonstrate the usefulness of our proposed method to analyze heteroscedasticity.