Several Researches On Quantile Regression And Robust Statistics

Composite quantile regression（CQR）is an efficient method to estimate parame-ters of the linear model with non-Gaussian random noise.The non-smoothness of CQR loss prevents many efficient algorithms from being used.Essentially,this is due to the non-smoothness of quantile regression（QR）loss function.Recently,the convolution-type smoothed quantile regression（SQR）model has been proposed to overcome such shortcoming,and researchers confirmed that the SQR estimator is as good as the QR estimator in theory.In the first part of this paper,we propose the composite smoothed quantile regression（CSQR）model and investigate the inference problem for a large-scale dataset,in which the dimensionality p is allowed to increase with the sample size n while p/n∈（0,1）.After applying the convolution smoothing technique to the com-posite quantile loss,we obtain the convex and twice differentiable CSQR loss function,which can be optimized via the gradient descent algorithm.Theoretically,we estab-lish the non-asymptotic error bound for the CSQR estimators,and further provide the Bahadur representation and the Berry-Esseen bound,from which the asymptotic nor-mality of CSQR estimator can be immediately derived.To make valid inference,we construct the confidence intervals that based on the asymptotic distribution.Besides,we also explore the asymptotic relative efficiency of the CSQR estimator with respect to the standard composite quantile regression（CQR）estimator.At the end of this part,we provide extensive numerical experiments on both simulated and real data to demonstrate the good performance of our CSQR estimator compared to some baselines.l₁-penalized quantile regression（l₁-QR）is a useful tool for modeling the relation-ship between input and output variables when detecting heterogeneous effects in the high-dimensional setting.Hypothesis tests can then be formulated based on the debi-asedl₁-QR estimator that reduces the bias induced by Lasso penalty.However,the non-smoothness of the quantile loss brings great challenges to the computation,especially when the data dimension is ultra-high.After noting the success of the SQR model in both fixed and increasing dimensional scenarios,people also applied convolution-type smoothing technique tol₁-QR model,proposedl₁-SQR model and developed theory of estimation and variable selection therein.It has been verified thatl₁-SQR estimator achieves the same error rate as that ofl₁-QR estimator.In the second part of this work,we combine the debiased method with SQR model and come up with the debiasedl₁-SQR estimator,based on which we then establish confidence intervals and hypothesis testing in the high-dimensional setup.Theoretically,we provide the non-asymptotic Bahadur representation for our proposed estimator and also the Berry-Esseen bound,which implies the empirical coverage rates for the studentized confidence intervals.Furthermore,we build up the theory of hypothesis testing on both a single variable and a group of variables from the viewpoint of minimax theory.Finally,we exhibit exten-sive numerical experiments on both simulated and real data to demonstrate the good performance of our method.In the last part of this article,we propose a class of estimators that are based on the robust and computationally efficient gradient estimation for both low and high di-mensional risk minimization framework.The gradient estimation used in this work is co-ordinately built up by a series of newly raised univariate robust-and-efficient mean estimators.Our proposed estimators are iteratively from a variant of gradient descent method,in which the update direction is decided by the above robust and computation-ally efficient gradient.These estimators are not only of explicit expression and can be directly obtained via arithmetic,but also they are robust with respect to the presence of arbitrary outliers for common statistical models.Theoretically we ensure the conver-gence of the algorithms and achieve the non-asymptotic error bounds of these iterative estimators.Specifically,we apply our method to linear regression model and logistic regression model,and we attain robust parameter estimations and the corresponding ex-cess risk bounds.At last,we provide extensive simulation experiments on both low and high dimensional linear models to demonstrate the better performance of our proposed estimators compared to some baselines.