Weighted ℓ1ℓ1-penalized corrected quantile regression for high dimensional measurement error models

Standard formulations of prediction problems in high dimension regression models assume the availability of fully observed covariates and sub-Gaussian and homogeneous model errors. This makes these methods inapplicable to measurement errors models where covariates are unobservable and observations are possibly non sub-Gaussian and heterogeneous. We propose a weighted penalized corrected quantile estimator for regression parameters in linear regression models with additive measurement errors, where unobservable covariate is nonrandom. The proposed estimators forgo the need for the above mentioned model assumptions. We study these estimators in a high dimensional sparse setup where the dimensionality can grow exponentially with the sample size. We provide bounds for the statistical error associated with the estimation, that hold with asymptotic probability 1, thereby providing the ℓ1ℓ1-consistency of the proposed estimator. We also establish the model selection consistency in terms of the correctly estimated zero components of the parameter vector. A simulation study that investigates the finite sample accuracy of the proposed estimator is also included in the paper.