  Regression with the lasso penalty is a popular tool for performing dimension
reduction when the number of covariates is large. In many applications of the
lasso, like in genomics, covariates are subject to measurement error. We study
the impact of measurement error on linear regression with the lasso penalty,
both analytically and in simulation experiments. A simple method of correction
for measurement error in the lasso is then considered. In the large sample
limit, the corrected lasso yields sign consistent covariate selection under
conditions very similar to the lasso with perfect measurements, whereas the
uncorrected lasso requires much more stringent conditions on the covariance
structure of the data. Finally, we suggest methods to correct for measurement
error in generalized linear models with the lasso penalty, which we study
empirically in simulation experiments with logistic regression, and also apply
to a classification problem with microarray data. We see that the corrected
lasso selects less false positives than the standard lasso, at a similar level
of true positives. The corrected lasso can therefore be used to obtain more
conservative covariate selection in genomic analysis.
