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

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 an existing entropy-based test using simulation. It is observed that the proposed test performs well. Finally, some real data sets are analysed for illustrative purposes.

💡 Research Summary

This paper addresses the statistical estimation of Tsallis entropy, a one‑parameter generalization of Shannon entropy, and demonstrates its utility in goodness‑of‑fit (GOF) testing. Starting from the classic Vasicek (1976) spacing‑based estimator for Shannon entropy, the authors extend the idea to Tsallis entropy by introducing four non‑parametric estimators that differ in the weighting scheme applied to m‑spacings: the original Vasicek‑type ((T_{\alpha}^{V})), an m‑spacing version ((T_{\alpha}^{H})), an Ebrahimi‑type ((T_{\alpha}^{E})), and a Noughabi‑Arghami‑type ((T_{\alpha}^{W})). Each estimator is expressed as a function of ordered sample differences ((X_{(i+m)}-X_{(i-m)})) scaled by the window size (m). Under the asymptotic regime (n\to\infty), (m\to\infty) and (m/n\to0), all four estimators are proved to be consistent for the true Tsallis entropy (T_{\alpha}(X)); when (\alpha\to1) they reduce to the Vasicek estimator of Shannon entropy.

The paper then proposes a spacing‑based estimator for the Tsallis divergence between an empirical distribution (F) and a parametric model (F_{\theta}). By substituting the Vasicek‑type density estimator (\hat f) into the divergence definition, the authors obtain a statistic (bT_{\alpha}(F,F_{\hat\theta})) that is shown to be distribution‑free (Lemma 3.1). Consequently, critical values can be derived from the empirical distribution of the statistic without recourse to parametric bootstrapping, simplifying GOF testing.

A major contribution is the adaptation of the spacing methodology to progressively Type‑II censored samples. The authors construct a censored‑data estimator by recomputing m‑spacings after each censoring step, preserving the asymptotic properties while controlling bias. Monte‑Carlo experiments demonstrate that the censored estimator’s mean‑squared error remains comparable to the uncensored case even under moderate to heavy censoring.

In addition to spacing‑based approaches, the paper introduces a quantile‑function‑based estimator. By differentiating the quantile function (Q(p)=F^{-1}(p)) to obtain the quantile density (q(p)), the authors define (\tilde T_{\alpha}) which relies on kernel‑smoothed estimates of (q(p)). They prove consistency and asymptotic normality (Theorem 6.2), and simulations show that (\tilde T_{\alpha}) achieves lower variance than kernel‑based entropy estimators while being computationally cheaper.

Using the divergence estimator, the authors develop GOF tests for normality and exponentiality. The test statistic is simply the estimated Tsallis divergence; larger values indicate departure from the null model. For exponentiality, a version that works with progressively censored data is also presented. Extensive simulation studies compare the proposed tests with classical procedures (Kolmogorov–Smirnov, Anderson–Darling, Shapiro–Wilk, etc.) and with recent entropy‑based tests. Results indicate that the Tsallis‑based tests have higher power, especially against alternatives that affect tail behavior, and maintain nominal type‑I error rates. In the censored exponentiality setting, the new test outperforms an existing entropy‑based censored test, showing lower false‑positive rates and higher detection power.

Tables 1 and 2 provide bias and mean‑squared error for the four spacing estimators across a range of sample sizes, window parameters, and α values for both normal and exponential distributions. The (T_{\alpha}^{W}) estimator consistently exhibits the smallest bias and MSE, and it is the most robust to contamination (e.g., 5 % Cauchy outliers). The authors also present real‑data analyses: (i) a biometric dataset to assess normality, and (ii) a lifetime dataset with progressive censoring to test exponentiality. In both cases the Tsallis‑based procedures yield clearer, more consistent p‑values than traditional methods.

Overall, the paper makes several substantive contributions: (1) four novel spacing‑based estimators for Tsallis entropy with proven asymptotic properties; (2) a distribution‑free estimator for Tsallis divergence enabling straightforward GOF testing; (3) extensions to progressive Type‑II censoring; (4) a quantile‑density based estimator with asymptotic normality; and (5) comprehensive simulation and real‑data evidence of superior performance. Limitations include the need for a data‑driven rule for selecting the window size (m) and the lack of discussion on multivariate extensions. Future work could explore adaptive bandwidth selection, high‑dimensional settings, and applications to dependent or heavy‑tailed data.