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Variance estimation based on blocked 3×2 cross-validation in high-dimensional linear regression.

Xingli Yang1, Yu Wang2, Wennan Yan1

  • 1School of Mathematical Sciences, Shanxi University, Taiyuan, People's Republic of China.

Journal of Applied Statistics
|June 16, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces a novel model selection method for high-dimensional linear regression to improve variance estimation. The proposed approach offers a higher probability of selecting the true model, reducing bias and variance compared to existing refitted cross-validation (RCV) methods.

Keywords:
High-dimensional linear regressionasymptotic normality propertyblocked 3×2 cross-validationvariance estimation

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Area of Science:

  • Statistics
  • Machine Learning
  • Econometrics

Background:

  • High-dimensional linear regression (where variable dimension exceeds sample size) presents challenges for traditional variance estimation.
  • Ordinary least squares (OLS) based variance estimation suffers from high bias due to spurious correlations between noise and predictors, even with sparsity.
  • Existing refitted cross-validation (RCV) methods may fail to select the true model in complex scenarios, leading to biased variance estimates.

Purpose of the Study:

  • To develop a robust model selection method for high-dimensional linear regression to improve variance estimation accuracy.
  • To address the limitations of RCV in selecting the true model, particularly in finite samples and complex settings.
  • To propose a novel method that yields lower bias and variance in variance estimation.

Main Methods:

  • A new model selection technique is proposed, utilizing ranks based on the frequency of occurrences from a blocked 3×2 cross-validation with six votes.
  • This method aims to enhance the probability of identifying the true model compared to the RCV approach.
  • The performance of the proposed method is evaluated through theoretical analysis and practical considerations.

Main Results:

  • The proposed model selection method demonstrates a considerably higher probability of including the true model in practice than the RCV method.
  • Variance estimation derived from the proposed method's selected model exhibits reduced bias and smaller variance.
  • Theoretical analysis confirms the asymptotic normality property of the proposed variance estimation.

Conclusions:

  • The novel blocked cross-validation approach offers a superior model selection strategy for high-dimensional linear regression.
  • This method effectively mitigates the bias and variance issues in traditional and RCV-based variance estimation.
  • The proposed technique provides a statistically sound and practically advantageous solution for accurate variance estimation in high-dimensional settings.