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This paper studies inference in the high-dimensional linear regression modelwith outliers. Sparsity constraints are imposed on the vector of coefficientsof the covariates. The number of outliers can grow with the sample size whiletheir proportion goes to 0. We propose a two-step procedure for inference onthe coefficients of a fixed subset of regressors. The first step is a based onseveral square-root lasso l1-norm penalized estimators, while the second stepis the ordinary least squares estimator applied to a well chosen regression. Weestablish asymptotic normality of the two-step estimator. The proposedprocedure is efficient in the sense that it attains the semiparametricefficiency bound when applied to the model without outliers underhomoscedasticity. This approach is also computationally advantageous, itamounts to solving a finite number of convex optimization programs.

High-dimensional inference robust to outliers with l1-norm penalization