An $\{\ell_{1},\ell_{2},\ell_{\infty}\}$-regularization approach to high-dimensional errors-in-variables models

Abstract

Several new estimation methods have been recently proposed for the linear regression model with observation errors in the design. Different assumptions on the data generating process have motivated different estimators and analysis. In particular, the literature considered (1) observation errors in the design uniformly bounded by some $\bar{\delta}$, and (2) zero-mean independent observation errors. Under the first assumption, the rates of convergence of the proposed estimators depend explicitly on $\bar{\delta}$, while the second assumption has been essentially applied when an estimator for the second moment of the observational error is available. This work proposes and studies two new estimators which, compared to other procedures for regression models with errors in the design, exploit an additional $\ell_{\infty}$-norm regularization. The first estimator is applicable when both (1) and (2) hold but does not require an estimator for the second moment of the observational error. The second estimator is applicable under (2) and requires an estimator for the second moment of the observation error. Importantly, we impose no assumption on the accuracy of this pilot estimator, in contrast to the previously known procedures. As the recent proposals, we allow the number of covariates to be much larger than the sample size. We establish the rates of convergence of the estimators and compare them with the bounds obtained for related estimators in the literature. These comparisons show interesting insights on the interplay of the assumptions and the achievable rates of convergence.