Pivotal Estimation Via Self-Normalization for High-Dimensional Linear Models with Errors in Variables

We propose a new estimator for the high-dimensional linear regression model with measurement error in the design where the number of coefficients is potentially larger than the sample size. The main novelty of our procedure is that the choice of penalty parameters is pivotal. The estimator is based on applying a self-normalization to the constraints that characterize the estimator. Importantly, we show how to cast the computation of the estimator as the solution of a convex program with second order cone constraints. This allows the use of algorithms with theoretical guarantees and enables reliable implementation. Under sparsity assumptions, we derive lq-rates of convergence and show that consistency can be achieved even if the number of regressors exceeds the sample size. We further provide a simple thresholded estimator that yields a provably sparse estimator with similar l2 and l1-rates of convergence.