Weighted l1-Penalized Corrected Quantile Regression for High-Dimensional Temporally Dependent Measurement Errors

Abstract

This article derives some large sample properties of weighted $l_1$-penalized corrected quantile estimators of the regression parameter vector in a high-dimensional errors in variables (EIVs) linear regression model. In this model, the number of predictors $p$ depends on the sample size $n$ and tends to infinity, generally at a faster rate than $n$, as $n$ tends to infinity. Moreover, the measurement errors in the covariates are assumed to have linear stationary temporal dependence and known Laplace marginal distribution while the regression errors are assumed to be independent non-identically distributed random variables having possibly heavy tails. The article discusses some rates of consistency of these estimators, a model consistency result and an appropriate data adaptive algorithm for obtaining a suitable choice of weights. A simulation study assesses the finite sample performance of some of the proposed estimators. This article also contains analogs of Massart's inequality for independent and short memory moving average predictors, which is instrumental in establishing the said consistency rates of the above mentioned estimators in the current setup of high dimensional EIVs regression models.