  Several new estimation methods have been recently proposed for the linear
regression model with observation error 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 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 $l_{\infty}$-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.
