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Abstract:
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We study the problem of robust estimation for the regression parameter in high-dimensional sparse heteroscedastic linear models. In our model, the variance of the random error in each observation depends on both predictor variables and regression coefficients. Our first goal is to provide theoretical guarantees of the Lasso under the assumed type of heteroscedasticity. We show in both theory and experiments that the Lasso is estimation consistent and variable selection consistent under mild sufficient conditions, even in the presence of heteroscedastic noise. The second goal of this work is to robustify the Lasso estimates given that the parametric form of the variance function is known. We propose a new one-step estimator and derive its asymptotical properties for uncertainty quantification and valid statistical inference. Our approach creates an efficient paradigm to treat heteroscedasticity for high-dimensional sparse linear models.
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