The Study Of Single-index Composite Quantile Regression For High-dimensional Data

Composite quantile regression（CQR）is becoming more popular due to its robustness and efficientce.It is showed that the relative efficiency of the CQR estimator compared with the least squares is greater than 70% regardless of the error distribution.In recent years,composite quantile regression has been widely used in the study of single-index models,and it has a wide range of applications in many scientific fields,such as biostatistics,economics and financial econometrics.Moreover,various fields have the characteristics of exponential growth of data scale.This thesis studies CQR method for single-index models with ultra-dimensional data.A new l1 regularzation based on the CQR method is introduced for effificiently estimating the index coeffificients in model and performing variable selection simultaneously.On the theoretical side,this thesis show that the proposed method enjoys an optimal statistical rate of convergence and converges to the true signal under mild conditions.Then,a new debiased CQR estimation for single-index models is developed by combining the debiasing technqiue with the CQR method.Thus,the asymptotic normality of the proposed estimation can be proven,which can then be used to constuct valid confidence intervals and hypothesis testing.Both simulations and data analysis are conducted to illustrate the finited sample performance of the proposed methods.