Rank-Based Sparse Regression in Principal Components Space under Measurement Error

2026-04-06

AI Summary

This study addresses the challenge of high-dimensional principal component regression when predictors are contaminated by additive measurement error and the response exhibits heavy-tailed errors or outliers. The authors propose a doubly robust estimation method that replaces the conventional squared loss with a Wilcoxon-type rank loss in the principal component space and incorporates a one-step adaptive reweighting strategy to correct the shrinkage bias of an initial ℓ₁-regularized estimator. This approach is the first to simultaneously achieve robustness against both heavy-tailed response errors and corrupted predictors. Theoretical analysis establishes prediction error bounds for the second-stage fitted mean under a fixed regularization parameter. Numerical experiments demonstrate that the method performs comparably to existing approaches under Gaussian noise yet exhibits markedly superior stability when heavy-tailed errors coexist with predictor contamination.

Abstract

We study high-dimensional regression in principal components space when the predictors are observed with additive measurement error and the response errors may be heavy-tailed. The starting point is the $\ell_1$-penalized principal-components estimator of Song and Zou (2026), which enjoys a blessing-of-dimensionality phenomenon under predictor contamination but senstive for heavy-tailed data or outliers. We replace the squared loss by a Wilcoxon-type rank loss and then apply a one-step adaptive reweighting scheme to reduce the shrinkage bias of the initial $\ell_1$ fit. The resulting procedure combines robustness to heavy-tailed response errors with the contamination geometry induced by the empirical principal-components basis. Our main theorem gives a prediction bound for the fixed-$λ$ second-stage fitted mean. Simulations show that the rank-based procedure is competitive under Gaussian noise and substantially more stable under heavy-tailed errors, especially when predictor contamination is present.

Problem

• measurement error
• heavy-tailed errors
• sparse regression
• principal components
• high-dimensional regression

Innovation

• rank-based regression
• measurement error
• principal components
• heavy-tailed errors
• adaptive reweighting