Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On testing the class of symmetry using entropy characterization and empirical likelihood approach

Ganesh Vishnu Avhad, Ananya Lahiri, Sudheesh K. Kattumannil

TL;DR

The authors develop an entropy-based characterization of symmetry using generalized cumulative entropy measures, establishing that symmetry is equivalent to $\mathcal{GCRE}(X)=\mathcal{GCE}(X)$ under suitable conditions. They construct a nonparametric symmetry test based on a U-statistic departure $\widehat{\Delta}_n$, and implement two Wilks-type inference procedures via jackknife empirical likelihood (JEL) and adjusted JEL (AJEL), with Monte Carlo critical values to avoid variance estimation burdens. Through extensive simulations, the proposed tests show favorable size control and higher power against skew alternatives compared with classical tests, across a range of sample sizes. Real-data applications demonstrate the methods’ practical utility for detecting symmetry or asymmetry in empirical distributions, with consistent conclusions across SCR, JEL, and AJEL analyses. Overall, the work provides robust, nonparametric tools for symmetry testing based on entropy concepts and resampling-based likelihood ratios, offering potential extensions to other information measures.

Abstract

In this paper, we obtain a new characterization result for symmetric distributions based on the entropy measure. Using the characterization, we propose a nonparametric test to test the symmetry of a distribution. We also develop the jackknife empirical likelihood and the adjusted jackknife empirical likelihood ratio tests. The asymptotic properties of the proposed test statistics are studied. We conduct extensive Monte Carlo simulation studies to assess the finite sample performance of the proposed tests. The simulation results indicate that the jackknife empirical likelihood and adjusted jackknife empirical likelihood ratio tests show better performance than the existing tests. Finally, two real data sets are analysed to illustrate the applicability of the proposed tests.

On testing the class of symmetry using entropy characterization and empirical likelihood approach

TL;DR

The authors develop an entropy-based characterization of symmetry using generalized cumulative entropy measures, establishing that symmetry is equivalent to under suitable conditions. They construct a nonparametric symmetry test based on a U-statistic departure , and implement two Wilks-type inference procedures via jackknife empirical likelihood (JEL) and adjusted JEL (AJEL), with Monte Carlo critical values to avoid variance estimation burdens. Through extensive simulations, the proposed tests show favorable size control and higher power against skew alternatives compared with classical tests, across a range of sample sizes. Real-data applications demonstrate the methods’ practical utility for detecting symmetry or asymmetry in empirical distributions, with consistent conclusions across SCR, JEL, and AJEL analyses. Overall, the work provides robust, nonparametric tools for symmetry testing based on entropy concepts and resampling-based likelihood ratios, offering potential extensions to other information measures.

Abstract

In this paper, we obtain a new characterization result for symmetric distributions based on the entropy measure. Using the characterization, we propose a nonparametric test to test the symmetry of a distribution. We also develop the jackknife empirical likelihood and the adjusted jackknife empirical likelihood ratio tests. The asymptotic properties of the proposed test statistics are studied. We conduct extensive Monte Carlo simulation studies to assess the finite sample performance of the proposed tests. The simulation results indicate that the jackknife empirical likelihood and adjusted jackknife empirical likelihood ratio tests show better performance than the existing tests. Finally, two real data sets are analysed to illustrate the applicability of the proposed tests.
Paper Structure (9 sections, 5 theorems, 68 equations, 2 figures, 6 tables, 1 algorithm)

This paper contains 9 sections, 5 theorems, 68 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Characterization result based on entropy measure
  3. Examples
  4. Test for symmetry
  5. Jackknife empirical likelihood ratio test
  6. Adjusted jackknife empirical likelihood ratio test
  7. Simulation study
  8. Data Analysis
  9. Conclusion

Key Result

Theorem 1

The following two statements are equivalent under the conditions $w(\cdot)$ is a positive weight function, for $x<u$, $\phi(\cdot)\geq \phi(u)$ and $x>u$, $\phi(\cdot)\leq \phi(u)$: $(i)$$X$ is a symmetric random variable; $(ii)$$\mathcal{GCRE}(X)=\mathcal{GCE}(X)$.

Figures (2)

  • Figure 1: Modeling tensile strength data using normal distribution
  • Figure 2: Boxplot

Theorems & Definitions (10)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Theorem 3
  • proof
  • Theorem 4
  • proof