Optimal Bias-Correction and Valid Inference in High-Dimensional Ridge Regression: A Closed-Form Solution

• URL: http://arxiv.org/abs/2405.00424v2
• Date: Wed, 24 Jul 2024 15:59:56 GMT
• Title: Optimal Bias-Correction and Valid Inference in High-Dimensional Ridge Regression: A Closed-Form Solution
• Authors: Zhaoxing Gao, Ruey S. Tsay,
• Abstract summary: We introduce an iterative strategy to correct bias effectively when the dimension $p$ is less than the sample size $n$. For $p>n$, our method optimally mitigates the bias such that any remaining bias in the proposed de-biased estimator is unattainable. Our method offers a transformative solution to the bias challenge in ridge regression inferences across various disciplines.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Ridge regression is an indispensable tool in big data analysis. Yet its inherent bias poses a significant and longstanding challenge, compromising both statistical efficiency and scalability across various applications. To tackle this critical issue, we introduce an iterative strategy to correct bias effectively when the dimension $p$ is less than the sample size $n$. For $p>n$, our method optimally mitigates the bias such that any remaining bias in the proposed de-biased estimator is unattainable through linear transformations of the response data. To address the remaining bias when $p>n$, we employ a Ridge-Screening (RS) method, producing a reduced model suitable for bias correction. Crucially, under certain conditions, the true model is nested within our selected one, highlighting RS as a novel variable selection approach. Through rigorous analysis, we establish the asymptotic properties and valid inferences of our de-biased ridge estimators for both $p<n$ and $p>n$, where, both $p$ and $n$ may increase towards infinity, along with the number of iterations. We further validate these results using simulated and real-world data examples. Our method offers a transformative solution to the bias challenge in ridge regression inferences across various disciplines.

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