Smoothing Quantile Regression With Elastic Net Penalty

High-dimensional data are commonly encountered in various scientific fields,such as information technology,biology,economics and so on.This poses great challenges to modern statistical analysis and optimal computation.And the traditional regression methods can not have good performance.In this paper,considering the fact that the high dimensionality often induces the collinearity problem and the error of high-dimensional data may be heavy-tailed,the traditional linear regression method can not deal with such data,hence we introduce the penalized quantile regression with the elastic-net that combines the strengths of the quadratic regularization and the lasso shrinkage.In addi-tion,the elastic net encourages a grouping effect,where strongly correlated predictors tend to be in or out of the model together.The quantile loss function is convex but non-smooth,so it is not easy to solve the model.Hence,by smoothing quantile loss function with the Huber smooth function,we give the smoothing quantile regression with elastic net penalty(SQEN).In addition,we derive the statistical consistent prop-erty of the SQEN estimator.To make the SQEN practically feasible,we propose an efficient iterative SQEN-MM method and establish its global convergence.Numerical results are reported to demonstrate the efficiency of our proposed method.This paper is divided into six chapters.In Chapter 1,we introduce the research background and the research status.In Chapter 2,we introduce the smoothing quantile regression with the elastic net penalty(SQEN).In Chapter 3,we derive the statistical properties of the SQEN estimator.In Chapter 4,we establish the algorithm of the SQEN-MM and show its global convergence.In Chapter 5,numerical experiments are reported to show the efficiency of the proposed method.Finally,we make some conclusions and prospect the future work of further research.