Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Estimation of Tsallis entropy and its applications to goodness-of-fit tests

Siddhartha Chakraborty, Asok K. Nanda, Narayanaswamy Balakrishnan

TL;DR

This work tackles the problem of estimating Tsallis entropy and Tsallis-based divergences from data and applying these measures to goodness-of-fit testing. It introduces four nonparametric estimators for Tsallis entropy based on m-spacings ($T_{\alpha}V$, $T_{\alpha}H$, $T_{\alpha}E$, $T_{\alpha}W$), a Tsallis-divergence estimator $\widehat{T}_{\alpha}(F;F_{\hat{\theta}})$, a Vasicek-type estimator for Tsallis entropy under progressive type-II censoring, and a quantile-density based estimator $T^q_{\alpha}(X)$ with asymptotic properties. The authors develop goodness-of-fit tests for normal and exponential distributions, extend methods to progressive Type-II samples, and demonstrate performance through simulations and real data analyses, often showing improvements over classical entropy-based tests while remaining robust to censoring. The proposed framework provides practical tools for model checking and reliability analyses, with clear guidance on estimator choice and censoring scenarios. Overall, the paper contributes a comprehensive, nonparametric toolkit for Tsallis entropy estimation and hypothesis testing in both complete and censored data settings.

Abstract

In this paper, we consider the problem of estimating Tsallis entropy from a given data set. We propose four different estimators for Tsallis entropy measure based on higher-order sample spacings, and then discuss estimation of Tsallis divergence measure. We compare the performance of the proposed estimators by means of bias and mean squared error and also examine their robustness to outliers. Next, we propose a spacings-based estimator for Tsallis entropy under progressive type-II censoring and study its performance using Monte Carlo simulations. Another estimator for Tsallis entropy is proposed using quantile function and its consistency and asymptotic normality are studied, and its performance is evaluated through Monte Carlo simulations. Goodness-of-fit tests for normal and exponential distributions as applications are developed using Tsallis divergence measure. The performance of the proposed tests are then compared with some known tests using simulations and it is shown that the proposed tests perform very well. Also, an exponentiality test under progressive type-II censoring is proposed, its performance is compared with existing entropy-based tests using simulation. It is observed that the proposed test performs well. Finally, some real data sets are analysed for illustrative purposes.

Estimation of Tsallis entropy and its applications to goodness-of-fit tests

TL;DR

This work tackles the problem of estimating Tsallis entropy and Tsallis-based divergences from data and applying these measures to goodness-of-fit testing. It introduces four nonparametric estimators for Tsallis entropy based on m-spacings (, , , ), a Tsallis-divergence estimator , a Vasicek-type estimator for Tsallis entropy under progressive type-II censoring, and a quantile-density based estimator with asymptotic properties. The authors develop goodness-of-fit tests for normal and exponential distributions, extend methods to progressive Type-II samples, and demonstrate performance through simulations and real data analyses, often showing improvements over classical entropy-based tests while remaining robust to censoring. The proposed framework provides practical tools for model checking and reliability analyses, with clear guidance on estimator choice and censoring scenarios. Overall, the paper contributes a comprehensive, nonparametric toolkit for Tsallis entropy estimation and hypothesis testing in both complete and censored data settings.

Abstract

In this paper, we consider the problem of estimating Tsallis entropy from a given data set. We propose four different estimators for Tsallis entropy measure based on higher-order sample spacings, and then discuss estimation of Tsallis divergence measure. We compare the performance of the proposed estimators by means of bias and mean squared error and also examine their robustness to outliers. Next, we propose a spacings-based estimator for Tsallis entropy under progressive type-II censoring and study its performance using Monte Carlo simulations. Another estimator for Tsallis entropy is proposed using quantile function and its consistency and asymptotic normality are studied, and its performance is evaluated through Monte Carlo simulations. Goodness-of-fit tests for normal and exponential distributions as applications are developed using Tsallis divergence measure. The performance of the proposed tests are then compared with some known tests using simulations and it is shown that the proposed tests perform very well. Also, an exponentiality test under progressive type-II censoring is proposed, its performance is compared with existing entropy-based tests using simulation. It is observed that the proposed test performs well. Finally, some real data sets are analysed for illustrative purposes.
Paper Structure (16 sections, 6 theorems, 55 equations, 4 figures, 14 tables)

This paper contains 16 sections, 6 theorems, 55 equations, 4 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Estimators for Tsallis entropy based on $m$-spacings
  3. Estimation of Tsallis divergence
  4. Comparison of estimators
  5. Robustness of the estimators in the presence of outliers
  6. Estimation of Tsallis entropy based on progressive type-II censored data using $m$-spacings
  7. Empirical evaluation
  8. Estimation of Tsallis entropy through quantile approach
  9. Empirical study
  10. Applications to Goodness-of-fit testing
  11. Test for Normality
  12. Test for Exponentiality
  13. Goodness-of-fit Tests for Progressive Type-II Samples
  14. Exponentiality under Progressive Type-II Samples
  15. Data Analyses
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $X_{1},\cdots,X_{n}$ be a random sample from a continuous distribution with cdf $F$ and pdf $f$. Then, as $m,n\to \infty$ and $\frac{m}{n}\to 0$, the estimators $T_{\alpha}V$, $T_{\alpha}H$, $T_{\alpha}E$ and $T_{\alpha}W$ all converge to the true value $T_{\alpha}(X)$.

Figures (4)

  • Figure 1: Empirical influence function for $T_{\alpha}V$.
  • Figure 2: Empirical influence function for $T_{\alpha}E$.
  • Figure 3: Histogram of $T^q_{\alpha}(X)-T_{\alpha}(X)$ for N(0,1).
  • Figure 4: Histogram of $T^q_{\alpha}(X)-T_{\alpha}(X)$ for Exp(1).

Theorems & Definitions (12)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Theorem 6.1
  • proof
  • Theorem 6.2
  • ...and 2 more