  In many problems involving generalized linear models, the covariates are
subject to measurement error. When the number of covariates p exceeds the
sample size n, regularized methods like the lasso or Dantzig selector are
required. Several recent papers have studied methods which correct for
measurement error in the lasso or Dantzig selector for linear models in the p>n
setting. We study a correction for generalized linear models based on Rosenbaum
and Tsybakov's matrix uncertainty selector. By not requiring an estimate of the
measurement error covariance matrix, this generalized matrix uncertainty
selector has a great practical advantage in problems involving high-dimensional
data. We further derive an alternative method based on the lasso, and develop
efficient algorithms for both methods. In our simulation studies of logistic
and Poisson regression with measurement error, the proposed methods outperform
the standard lasso and Dantzig selector with respect to covariate selection, by
reducing the number of false positives considerably. We also consider
classification of patients on the basis of gene expression data with noisy
measurements.
