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Variable Selection and Regularization in Quantile Regression via Minimum Covariance Determinant Based Weights.

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Summary

This study introduces penalized weighted quantile regression to improve variable selection and regularization in multiple regression. It effectively addresses issues caused by high leverage points and collinearity influential points in predictor data.

Keywords:
LASSO penaltyRIDGE penaltycollinearity influential pointselastic net penaltyhigh leverage pointsminimum covariance determinantweighted quantile regression

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Area of Science:

  • Statistics
  • Data Science
  • Regression Analysis

Background:

  • Variable selection and regularization are crucial in multiple regression but are sensitive to data aberrations.
  • Existing robust methods like quantile regression handle response outliers but not predictor outliers (high leverage points).
  • High leverage points can distort the predictor matrix's eigen-structure, creating or masking collinearity (collinearity influential points).

Purpose of the Study:

  • To propose a novel penalized weighted quantile regression method.
  • To generalize penalized weighted least absolute deviation to all quantile levels.
  • To provide a remedy against collinearity influential points and high leverage points in regression analysis.

Main Methods:

  • Generalization of penalized weighted least absolute deviation to penalized weighted quantile regression.
  • Incorporation of RIDGE, LASSO, and elastic net penalties.
  • Utilization of robust weights derived from the minimum covariance determinant (MCD) estimator for enhanced robustness.

Main Results:

  • The proposed method demonstrates improved robustness against high leverage points and collinearity influential points.
  • Simulations and applications show enhanced performance in variable selection and regularization compared to existing methods.
  • The robust weighting formulation is key to the observed improvements in regression analysis.

Conclusions:

  • Penalized weighted quantile regression offers a robust solution for variable selection and regularization in the presence of predictor space outliers.
  • The method effectively mitigates the adverse effects of high leverage and collinearity influential points.
  • This approach enhances the reliability and accuracy of multiple regression analyses.