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Empirical Likelihood Based Inference for a Divergence Measure Based on Survival Extropy

Naresh Garg, Isha Dewan, Sudheesh Kumar Kattumannil

TL;DR

This work proposes a survival-based divergence measure $D$ built from squared differences of survival functions, grounded in extropy, and develops three nonparametric estimators (U-statistics, empirical, kernel). It provides two inference schemes—normal approximation with jackknife variance and jackknife empirical likelihood—for constructing confidence intervals, with a comprehensive Monte Carlo study showing the U-statistic estimator often delivers the best accuracy and JEL CIs outperforming normal methods. The authors demonstrate practical utility through real-data analyses in biomedical survival studies and image-dbased tasks, illustrating how $D$ can detect subtle differences between populations and even small image changes. Overall, the paper advances flexible, nonparametric tools for comparing lifetime distributions and demonstrates their applicability in diverse domains.

Abstract

Survival extropy, which quantifies the uncertainty associated with the remaining lifetime distribution, provides an information-theoretic perspective on survival behavior. We consider a divergence measure based on survival extropy and derive its nonparametric estimators based on U-statistics, empirical distribution functions, and kernel density. Further, we construct confidence intervals for the divergence measure using the jackknife empirical likelihood (JEL) method and the normal approximation method with a jackknife pseudo-value-based variance estimator. A comprehensive Monte Carlo simulation study is conducted to compare the performance of the measure with existing divergence measures. Additionally, we evaluate the finite-sample performance of various estimators for the proposed measure. The findings highlight the effectiveness of the divergence measure and its estimators in practical applications. Finally, we show how the proposed divergence measure is used to detect the small differences between images in image datasets.

Empirical Likelihood Based Inference for a Divergence Measure Based on Survival Extropy

TL;DR

This work proposes a survival-based divergence measure built from squared differences of survival functions, grounded in extropy, and develops three nonparametric estimators (U-statistics, empirical, kernel). It provides two inference schemes—normal approximation with jackknife variance and jackknife empirical likelihood—for constructing confidence intervals, with a comprehensive Monte Carlo study showing the U-statistic estimator often delivers the best accuracy and JEL CIs outperforming normal methods. The authors demonstrate practical utility through real-data analyses in biomedical survival studies and image-dbased tasks, illustrating how can detect subtle differences between populations and even small image changes. Overall, the paper advances flexible, nonparametric tools for comparing lifetime distributions and demonstrates their applicability in diverse domains.

Abstract

Survival extropy, which quantifies the uncertainty associated with the remaining lifetime distribution, provides an information-theoretic perspective on survival behavior. We consider a divergence measure based on survival extropy and derive its nonparametric estimators based on U-statistics, empirical distribution functions, and kernel density. Further, we construct confidence intervals for the divergence measure using the jackknife empirical likelihood (JEL) method and the normal approximation method with a jackknife pseudo-value-based variance estimator. A comprehensive Monte Carlo simulation study is conducted to compare the performance of the measure with existing divergence measures. Additionally, we evaluate the finite-sample performance of various estimators for the proposed measure. The findings highlight the effectiveness of the divergence measure and its estimators in practical applications. Finally, we show how the proposed divergence measure is used to detect the small differences between images in image datasets.

Paper Structure

This paper contains 15 sections, 40 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Estimation of $D$
  3. U-statistics-based estimator
  4. Empirical distribution function-based estimator
  5. Kernel-based estimator
  6. Confidence interval for $D$
  7. Normal approximation with Jackknife pseudo-values for C.I. of $D$
  8. Jackknife empirical likelihood (JEL) based C.I. of $D$
  9. Monte Carlo Simulations
  10. Comparison of MSEs for $D$ and estimators of existing measures
  11. Comparison of estimators for $D$
  12. Comparison of confidence intervals for $D$
  13. Real Data Analysis
  14. Application on Image Datasets
  15. Concluding Remarks

Figures (6)

  • Figure 1: Empirical survival curves for comparing the survival functions of the standard and test treatment groups in the Veteran dataset.
  • Figure 2: Empirical survival curves for comparing the survival functions of male and female patients in the Lung Cancer dataset.
  • Figure 3: Empirical survival curves for comparing the survival functions of male and female patients in the PBC dataset.
  • Figure 4: Empirical survival curves for comparing the survival functions of patients receiving hormonal therapy versus no hormonal therapy in the GBSG2 dataset.
  • Figure 5: Axial and Coronal MRI Images of Brains Representing Different Tumor Categories
  • ...and 1 more figures