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.

翻译：本文研究了从给定数据集中估计Tsallis熵的问题。我们基于高阶样本间距提出了四种不同的Tsallis熵估计量，并讨论了Tsallis散度度量的估计。通过偏差和均方误差比较了所提估计量的性能，同时检验了其对异常值的稳健性。其次，我们提出了逐步II型删失下基于间距的Tsallis熵估计量，并通过蒙特卡洛模拟研究了其性能。利用分位数函数提出了另一种Tsallis熵估计量，研究了其相合性与渐近正态性，并通过蒙特卡洛模拟评估了其性能。作为应用，我们利用Tsallis散度度量发展了针对正态分布和指数分布的拟合优度检验。通过模拟将所提检验与若干已知检验进行比较，结果表明所提检验性能优异。此外，提出了逐步II型删失下的指数性检验，通过模拟将其与现有基于熵的检验进行比较，发现所提检验表现良好。最后，为说明方法的应用，分析了若干实际数据集。