A Comparative Analysis of Implementing Adaptive Lasso Penalty in Hierarchical Data: Quantile versus Mean Regression

This paper proposes a new penalized procedure for dealing with high-dimensional hierarchical data in the context of quantile regression models. The penalty used in our method is based on adaptive lasso. Hence, the current paper introduces a new optimization problem procedure to shrink the variables and simultaneously perform the variable selection task on the group level and regression coefficients. In addition, because heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges in modern statistical analysis, this paper investigates the application of the adaptive lasso in the quantile and mean regression models to treat hierarchical data. The simulation study shows that the adaptive lasso for the quantile regression with hierarchical data performs better than the adaptive lasso for the mean regression in both cases, considering the heavy-tailed distribution and some skewed densities for the error terms. In addition, the simulation study compared the adaptive lasso penalty with other popular penalties, and it was shown that the adaptive lasso performs better than others. The application of the proposed approach was also highlighted in real data analysis.