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Addressing both variable selection and misclassified responses with parametric and semiparametric methods

Hui Guo, Grace Y. Yi, Boyu Wang

Abstract

While variable selection has received extensive attention in the literature, its exploration in the presence of response measurement error remains underexplored. In this paper, we investigate this important problem within the context of binary classification with error-prone responses. We present valid variable selection procedures to address the complexities of response errors. Leveraging validation data, we introduce both parametric and semiparametric methodologies to accommodate the mismeasurement effects. By rigorously establishing theoretical results, we offer insights and justifications of the validity of the proposed methods. By properly choosing {the} penalty function and regularization parameter, we demonstrate that the resulting estimators possess the oracle property. To assess the finite sample properties of the proposed methods, we conduct numerical studies that confirm the effectiveness of our proposed methods.

Addressing both variable selection and misclassified responses with parametric and semiparametric methods

Abstract

While variable selection has received extensive attention in the literature, its exploration in the presence of response measurement error remains underexplored. In this paper, we investigate this important problem within the context of binary classification with error-prone responses. We present valid variable selection procedures to address the complexities of response errors. Leveraging validation data, we introduce both parametric and semiparametric methodologies to accommodate the mismeasurement effects. By rigorously establishing theoretical results, we offer insights and justifications of the validity of the proposed methods. By properly choosing {the} penalty function and regularization parameter, we demonstrate that the resulting estimators possess the oracle property. To assess the finite sample properties of the proposed methods, we conduct numerical studies that confirm the effectiveness of our proposed methods.
Paper Structure (20 sections, 4 theorems, 47 equations, 1 figure)

This paper contains 20 sections, 4 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and model setup
  3. Parametric method
  4. Observed likelihood function
  5. Penalized likelihood estimator
  6. Semiparametric method
  7. Implementation procedures
  8. Optimization algorithm
  9. Tuning parameters selection
  10. Implementation details
  11. Parametric method
  12. Semiparametric method
  13. Numerical studies
  14. Simulation study
  15. Simulation design
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose that the penalty function $\rho(\cdot)$ satisfies Condition B4-3 in Appendix B. Assume that the distributions of $\mathcal{D}_{\text{\tiny M}}$ and $\mathcal{D}_{\text{\tiny V}}$ satisfy the regularity conditions outlined in B1 of the Supplementary Material. If $b_n\rightarrow 0$ and $\lambd where $\|a\|_2\triangleq \sqrt{\sum_{j=1}^{r}a^2_j}$ represents the $L_2$ norm for vector $a\triang

Figures (1)

  • Figure 1: Boxplots of performance metrics for the NHANES obesity data, using the SCAD penalty, the GCV selector, and the logit link function for model (\ref{['Model']}).

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2: Oracle Property
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 3
  • Theorem 4: Oracle Property
  • Remark 4