Islam Abdullah Husseiny

This paper develops a general framework for Tsallis entropy (TsE) and cumulative residual Tsallis entropy (CRTsE) for concomitants of generalized order statistics under the Cambanis family of bivariate distributions. Explicit closed-form expressions and alternative representations for TsE and its cumulative residual counterparts, including a unique formulation of the alternative CRTsE, are derived. Various structural properties are established, including symmetry relations between low- and high-rank concomitants, as well as connections with classical submodels such as order statistics and record values. A nonparametric estimator for the CRTsE is proposed; its mean, variance, and consistency properties are investigated analytically and through extensive Monte Carlo simulations. The theoretical framework is further illustrated through two application domains: reliability assessment of real bivariate system data, and histogram-based image segmentation via a rank-sensitive Tsallis-thresholding scheme. The proposed methodology constitutes a flexible, dependence-aware generalization of classical Shannon-based entropy measures and is particularly advantageous for systems exhibiting nonlinear dependence, tail sensitivity, or non-extensive behavior.

Information measures in statistical distributions have wide-ranging applications in fields such as reliability engineering, information theory, hydrology, and environmental sciences, where understanding variability and risk is essential for decision making and system design. Motivated by this, we investigate and analyze uncertainty measures, namely, entropy and extropy measures, within the framework of the power Lomax distribution (PoLxD), a heavy-tailed model widely used in modeling lifetime and environmental data. We introduce novel formulations of these measures based on the quantile function of the PoLxD, offering a robust and insightful alternative to traditional methods. To evaluate the efficiency and reliability of the proposed estimators of these measures, we conducted an extensive simulation study, assessing performance through bias, mean squared error, and mean relative error. We examined and commented on the outcomes of calculating these measures, their relationship with the distribution’s parameters, and when they decreased and increased. Finally, we validate our approach by applying the proposed uncertainty measures to two real-world datasets where the PoLxD provided a superior fit.

Many disciplines, including environmental sciences, reliability engineering, hydrology, and information theory, rely heavily on information measures derived from statistical distributions for decision-making, risk assessment, and system design. Motivated by this need, the present study provides a comprehensive analysis of traditional and generalized information measures within the framework of the New X-Lindley distribution (NXLID). We systematically explore key concepts, including Fisher information, Shannon entropy, Tsallis entropy, Rényi entropy, extropy, and their cumulative and weighted counterparts, providing analytical expressions for each measure. Extensive simulation studies complement theoretical derivations to evaluate the performance of the proposed estimators in terms of bias, mean squared error, and mean relative error. To demonstrate the practical utility of the NXLID and its associated information measures, we analyze two real-world datasets related to component failure times in reliability contexts. In addition, we investigate the behavior of these information measures concerning order statistics, providing deeper insights into their distributional properties and variability. The findings underscore the flexibility and effectiveness of the NXLID in modeling lifetime data and assessing system reliability, establishing it as a valuable tool for both theoretical exploration and applied statistical modeling.

Thermodynamics, statistical mechanics, and quantum mechanics are the branches of physics that use entropy and extropy measures. It also seems to quantify the degree of signal uncertainty in information theory. Recently, extropy has been widely used in the literature as a measure of dispersion or uncertainty. Within generalized Farlie-Gumbel-Morgenstern families, the Cambanis bivariate family holds a prominent position as an efficient and adaptable extended family. Motivated by this, our work gives numerous characterizations of Cambanis copula based on order statistics (OSs). Also, we study the negative cumulative residual extropy and various measures of information associated with it for concomitants generalized OSs. Moreover, we study OSs and sequential OSs for each information measure in a computational study. This study offers a few beautiful symmetrical and asymmetric relationships that these information measures satisfy. Additionally, the empirical technique is applied to the Cambanis family to estimate every measure of negative extropy discussed in this work. As a final step, we have examined a bivariate real-world data set for illustrative purposes.

In this paper, the Fisher information (FI), relevant to m-generalized order statistics (m-GOSs) and their concomitants of the shape-parameter of the Sarmanov family of bivariate distributions, is investigated. In addition, we study the concomitants of m-GOSs from this family. Furthermore, we look at how those concomitants were distributed collectively. The FI contained in the scale and shape parameters of the exponential and power function distributions, respectively, in concomitants of m-GOSs is obtained. A study of the asymptotic behavior of the concomitants of ordinary order statistics is also provided. Some versatile applications for this study are offered. As a final step, we examined a bivariate real-world data set for illustrative purposes.