Balanced Penalized Quantile Regression With High-dimensional Heavy-tailed Data

As a result of the rapid development of modern computer technology,high-dimensional heavy-tailed data has become more and more important in the fields of data analysis such as:economics,finance,biology and medicine.Due to the sparsity of high-dimensional data,the resulting model would reduce not only interpretability but also precision of model if consid-er all the variables.So selecting variables at first is a nature thought when deal with high-dimensional data.Penalty function is an instrument which occupies a decisive position for these high-dimensional sparse models(HDSMs)in variable selection and it will result in parameter estimation simultaneously.Generally speaking,there 're mainly two penalty classes:convex penalty and concave penalty.The convex one,led by the least absolute shrinkage and selection operator(LASSO),will provide a well-defined global optimizer but can not screen the noisy variable precisely.On the contrast,the other one just has the opposite properties that it will select significant variables more precisely but may not result in a stable global operator when in high-dimensional situation.The complementarities give rise to the idea of the combination of them.Based on existing study,the convex penalty in the method make contributes to variable screening and the concave one to rectify the deviation and precisely variable selection.Unfortunately the method is under the ordinary least square.That means for heavy-tailed data the fitting effect will not good enough.Compared to ordinary least square,quantile regression has three advantages:(?)it has loos-er restriction about the distribution of data;(?)it can give a more comprehensive analysis about the relationship between variants and responses based on conditional quantiles;(?)the estima-tor has robustness.These advantages make it an important substitute to ordinary least square.To sum up,penalized quantile regression has an ignorable theoretical and realistic significance for dealing with high-dimensional heavy-tailed data.In this paper,we propose a new way for penalty method which applying convex and con-cave penalty to quantile regression simultaneously and study its properties.The conclusion will show that under some usual conditions,the balanced method make its two components play in different roles,i.e.,the convex penalty for variable screening and the concave one for rectifying the deviation and precisely variable selection.The study makes a good complement to penal-ized quantile regression.The numerical analysis indicates that compared to existing methods,the balanced one will obtain an overall estimator for different distributions,and have smaller deviation.Empirical analysis will also be given in this paper.