Rate Optimal Estimation and Confidence Intervals for High-dimensional Regression with Missing Covariates
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Although a majority of the theoretical literature in high-dimensional statistics has focused on settings which involve fully-observed data, settings with missing values and corruptions are common in practice. We consider the problems of estimation and of constructing component-wise confidence intervals in a sparse high-dimensional linear regression model when some covariates of the design matrix are missing completely at random. We analyze a variant of the Dantzig selector [9] for estimating the regression model and we use a de-biasing argument to construct component-wise confidence intervals. Our first main result is to establish upper bounds on the estimation error as a function of the model parameters (the sparsity level s, the expected fraction of observed covariates ρ_*, and a measure of the signal strength β^*_2). We find that even in an idealized setting where the covariates are assumed to be missing completely at random, somewhat surprisingly and in contrast to the fully-observed setting, there is a dichotomy in the dependence on model parameters and much faster rates are obtained if the covariance matrix of the random design is known. To study this issue further, our second main contribution is to provide lower bounds on the estimation error showing that this discrepancy in rates is unavoidable in a minimax sense. We then consider the problem of high-dimensional inference in the presence of missing data. We construct and analyze confidence intervals using a de-biased estimator. In the presence of missing data, inference is complicated by the fact that the de-biasing matrix is correlated with the pilot estimator and this necessitates the design of a new estimator and a novel analysis. We also complement our mathematical study with extensive simulations on synthetic and semi-synthetic data that show the accuracy of our asymptotic predictions for finite sample sizes.
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