A sparse approach for high-dimensional data with heavy-tailed noise

High-dimensional data have commonly emerged in diverse fields, such as economics, finance, genetics, medicine, machine learning, and so on. In this paper, we consider the sparse quantile regression problem of high-dimensional data with heavy-tailed noise, especially when the number of regressors is much larger than the sample size. We bring the spirit of Lp-norm support vector regression into quantile regression and propose a robust Lp-norm support vector quantile regression for high-dimensional data with heavy-tailed noise. The proposed method achieves robustness against heavy-tailed noise due to its use of the pinball loss function. Furthermore, Lp-norm support vector quantile regression ensures that the most representative variables are selected automatically by using a sparse parameter. We use a simulation study to test the variable selection performance of Lp-norm support vector quantile regression, where the number of explanatory variables greatly exceeds the sample size. The simulation study confirms that Lp-norm support vector quantile regression is not only robust against heavy-tailed noise but also selects representative variables. We further apply the proposed method to solve the variable selection problem of index construction, which also confirms the robustness and sparseness of Lp-norm support vector quantile regression.