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M-estimation in high-dimensional linear model.

Kai Wang1, Yanling Zhu1

  • 1School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, P.R. China.

Journal of Inequalities and Applications
|March 7, 2019
PubMed
Summary
This summary is machine-generated.

This study explores M-estimation for high-dimensional linear regression, demonstrating its robustness and good properties under specific assumptions. The M-estimation framework encompasses various regression techniques, including least absolute deviation and quantile regression.

Keywords:
High-dimensionalityM-estimationOracle propertyPenalized methodVariable selection

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Area of Science:

  • Statistics
  • Econometrics
  • Machine Learning

Background:

  • High-dimensional linear regression models present unique statistical challenges.
  • Existing estimation methods like least squares may lack robustness in high dimensions.
  • M-estimation offers a flexible framework for robust statistical inference.

Purpose of the Study:

  • To investigate the properties of M-estimators in high-dimensional linear regression.
  • To analyze the impact of local linear approximation penalty terms on M-estimation.
  • To demonstrate the robustness and effectiveness of the proposed M-estimation method.

Main Methods:

  • Developed an M-estimator for high-dimensional linear regression.
  • Incorporated a local linear approximation penalty term.
  • Utilized numerical simulations with appropriate algorithms to assess performance.

Main Results:

  • The proposed M-estimator demonstrates desirable statistical properties under stated assumptions.
  • The M-estimation framework unifies methods such as least absolute deviation, quantile regression, least squares, and Huber regression.
  • Numerical simulations confirmed the method's good robustness.

Conclusions:

  • The M-estimation method, particularly with local linear approximation penalties, is a robust approach for high-dimensional linear regression.
  • This framework provides a unified perspective on several established regression techniques.
  • The findings support the practical applicability of M-estimation in complex data scenarios.