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This paper provides an alternative to penalized estimators for estimation andvari- able selection in high dimensional linear regression models withmeasurement error or missing covariates. We propose estimation via biascorrected least squares after model selection. We show that by separating modelselection and estimation, it is possible to achieve an improved rate ofconvergence of the L2 estimation error compared to the rate sqrt{s log p/n}achieved by simultaneous estimation and variable selection methods such as L1penalized corrected least squares. If the correct model is selected with highprobability then the L2 rate of convergence for the proposed method is indeedthe oracle rate of sqrt{s/n}. Here s, p are the number of non zero parametersand the model dimension, respectively, and n is the sample size. Under verygeneral model selection criteria, the proposed method is computationallysimpler and statistically at least as efficient as the L1 penalized correctedleast squares method, performs model selection without the availability of thebias correction matrix, and is able to provide estimates with only a smallsub-block of the bias correction covariance matrix of order s x s in comparisonto the p x p correction matrix required for computation of the L1 penalizedversion. Furthermore we show that the model selection requirements are met by acorrelation screening type method and the L1 penalized corrected least squaresmethod. Also, the proposed methodology when applied to the estimation ofprecision matrices with missing observations, is seen to perform at least aswell as existing L1 penalty based methods. All results are supportedempirically by a simulation study.

Two Stage Non-penalized Corrected Least Squares for High Dimensional Linear Models with Measurement error or Missing Covariates