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Yanan Song1, Haohui Han1, Liya Fu1

  • 1School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China.

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This study introduces a new penalized method for quantile regression, offering robust variable selection and parameter estimation for high-dimensional longitudinal data. The approach effectively handles outliers and complex data structures.

Keywords:
convolution‐type smoothinghigh‐dimensionlongitudinal dataquantile regression

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Area of Science:

  • Statistics
  • Biostatistics
  • Econometrics

Background:

  • Quantile regression is a robust alternative to linear regression for statistical modeling.
  • High-dimensional longitudinal data presents challenges in variable selection and parameter estimation.
  • Existing methods may be sensitive to outliers and struggle with correlated data within subjects.

Purpose of the Study:

  • To develop a penalized weighted convolution-type smoothed method for variable selection and robust parameter estimation in high-dimensional quantile regression.
  • To address the challenges posed by outliers and within-subject correlation in longitudinal data.
  • To establish the theoretical properties and practical performance of the proposed method.

Main Methods:

  • Utilizing a twice-differentiable and smoothed loss function instead of the traditional check function.
  • Employing efficient gradient-based iterative algorithms for variable selection.
  • Implementing a two-step weighted estimation method to account for within-subject correlation.
  • Proving oracle properties under regularity conditions.

Main Results:

  • The proposed method achieves consistent variable selection even when the number of covariates exceeds the sample size.
  • Robust parameter estimation is achieved by circumventing the influence of outliers.
  • Enhanced accuracy in parameter estimation due to the incorporation of within-subject correlation.
  • Demonstrated performance through simulation studies and real-data applications.

Conclusions:

  • The penalized weighted convolution-type smoothed method provides a powerful tool for analyzing high-dimensional longitudinal data.
  • The method offers robustness to outliers and improved estimation accuracy.
  • It consistently selects relevant variables, making it suitable for complex datasets.