  This paper considers errors-in-variables models in a high-dimensional setting
where the number of covariates can be much larger than the sample size, and
there are only a small number of non-zero covariates. The presence of
measurement error in the covariates can result in severely biased parameter
estimates, and also affects the ability of penalized methods such as the lasso
to recover the true sparsity pattern. A new estimation procedure called
SIMSELEX (SIMulation-SELection-EXtrapolation) is proposed. This procedure
augments the traditional SIMEX approach with a variable selection step based on
the group lasso. The SIMSELEX estimator is shown to perform well in variable
selection, and has significantly lower estimation error than naive estimators
that ignore measurement error. SIMSELEX can be applied in a variety of
errors-in-variables settings, including linear models, generalized linear
models, and Cox survival models. It is furthermore shown how SIMSELEX can be
applied to spline-based regression models. SIMSELEX estimators are compared to
the corrected lasso and the conic programming estimator for a linear model, and
to the conditional scores lasso for a logistic regression model. Finally, the
method is used to analyze a microarray dataset that contains gene expression
measurements of favorable histology Wilms tumors.
