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Functional Linear Model with Zero-value Coefficient Function at Sub-regions.

Jianhui Zhou1, Nae-Yuh Wang1, Naisyin Wang1

  • 1Department of Statistics, University of Virginia, Charlottesville, VA 22904, U.S.A.Department of Medicine, Johns Hopkins University, Baltimore, MD 21287, U.S.A.Department of Statistics, University of Michigan, Ann Arbor, MI 48109, U.S.A.

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Summary
This summary is machine-generated.

We developed a two-stage shrinkage method to accurately estimate coefficient functions in functional linear regression, identifying zero-value regions and improving non-zero coefficient estimation. This approach enhances statistical precision and numerical performance.

Keywords:
B-spline basis functionfunctional linear regressiongroup smoothly clipped absolute deviation approachnull region

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Area of Science:

  • Statistics
  • Functional Data Analysis

Background:

  • Functional linear regression models are essential for analyzing data where predictors or responses are functions.
  • Estimating coefficient functions, especially when they are zero in certain regions, presents challenges in statistical modeling.
  • Accurate identification of these 'null regions' is crucial for reliable inference and model interpretability.

Purpose of the Study:

  • To propose a novel two-stage shrinkage method for estimating coefficient functions in functional linear regression.
  • To accurately identify sub-regions where the coefficient function is zero (null regions).
  • To perform estimation and inference on the nonparametrically estimated coefficient function without over-shrinking non-zero values.

Main Methods:

  • A two-stage approach combining the Dantzig selector and a group SCAD (Smoothly Clipped Absolute Deviation) penalty.
  • Stage one uses the Dantzig selector for initial identification of the null region.
  • Stage two employs group SCAD to refine the null region estimation and provide coefficient function estimation and inference.

Main Results:

  • The proposed estimator possesses the Oracle property, meaning it correctly identifies the null region with high probability.
  • It achieves the same asymptotic normality as known-null-region estimators for the non-null regions.
  • The refined estimator overcomes the underestimation of non-zero coefficients observed with the initial Dantzig estimator, showing superior numerical performance.

Conclusions:

  • The proposed two-stage shrinkage method effectively estimates coefficient functions in functional linear regression, accurately identifying null regions.
  • This method offers advantages in reducing parameter complexity and improving numerical stability compared to traditional approaches.
  • The approach is validated through simulation studies and applied to real-world data, demonstrating its practical utility in identifying associations over specific ranges.