Confidence Intervals For High-Dimensional Regression When Covariates Contain Errors

In the standard high-dimensional regression theory and applied research,it is assumed that the covariates are completely observed.However,this assump-tion is unrealistic in many practical applications.We often encounter situa-tions where the covariates have observation errors.Common examples include questionnaires,sensor network data and gene expression data(see Loh and Wainwright(2012)).However,the statistical inference of these models is very challenging,because in the research of these models,besides dealing with the high dimensionality of parameters,measurement errors need to be considered to avoid non-vanishing biases.In order to alleviate the influence of measuremen-t errors in high dimensions,some new estimation methods for high dimension regression models with observation errors have been proposed in recent years.However,these methods mainly focus on point estimation.Moreover,the es-timators obtained by the above methods are non-linear and non-explicit,and none of these sparse estimators has a tractable limit distribution.Therefore,the research on confidence interval of high-dimensional error-in-variables regression model is seldom involved.Based on the convex optimization idea of CoCoLasso(see Datta and H.Zou(2017),this paper constructs the de-biased estimator of parameters of high-dimensional regression model with noise by invert the cor-responding KKT condition of CoCoLasso optimization problem,we prove the asymptotic normality of the estimator under certain regularization conditions,and give an effective algorithm for constructing the corresponding confidence interval.Numerical results further demonstrate the performance of the method.