This study develops estimators for Cumulative Residual Extropy (CR Extropy) and Dynamic Cumulative Residual Extropy (DCR Extropy) of the equilibrium distribution of order $r$ of Lomax distribution under progressively Type-II censored data. Maximum likelihood and Bayesian estimators are obtained. The Bayesian estimators are evaluated based on LINEX loss functions. Non-informative and informative priors are considered for unknown parameters. By using Lindley's approximation and the Importance Sampling Methods, it is possible to approximate the closed-form expressions of the Bayesian estimators. The confidence intervals for the estimators are calculated using the normal approximation method and bootstrap algorithms. We have conducted simulation studies to evaluate how well the proposed estimators perform. Additionally, we have tested the practical utility of these estimators using a real data.

This paper deals with the dynamic survival past extropy as a measure of uncertainty in the past lifetime distributions. We introduce a quantile version of the extropy function in past lifetime. Various properties of the proposed measure are obtained. Additionally, some stochastic comparisons and bounds are derived and the performance of the dynamic survival past extropy of parallel and series system is studied as well.

It is interesting to learn that the complementary dual of the Shannon entropy measure exists and has some common properties. This new measure of uncertainty has been introduced by Lad et al. (2015) and is known as extropy. Although there are some mathematical analogies between the two measures, extropy typically has different uses and interpretations than entropy. Taking into account the importance of extropy measure, and its various generalizations, in the present communication, we consider and study Kullback-Leibler based "divergence-extropy" measure between the distribution of nth upper k-record and mth upper k-record values. Characterization problems for the proposed "divergence-extropy" measure have been studied. Further, some specific lifetime distributions used in lifetime testing, physical sciences, survival analysis and reliability engineering have been studied using the proposed "divergence-extropy" measure. At the end, we study the proposed "divergence-extropy" measure between the distribution of k-record value and order statistics.

In this paper, a new bivariate family of distributions is proposed by mixing up the new family of heavy-tailed distributions approach with the Farlie-Gumble-Morgenstern copula. The resultant family may be called the Farlie-Gumble-Morgenstern new family of heavy-tailed distributions. For the proposed family, some properties of the concomitants of generalized order statistics are obtained. The joint distribution of concomitants is also derived. Furthermore, for this new family, some properties of extropy for the concomitant of generalized order statistics are obtained. The method of maximum likelihood estimation is implemented to obtain the estimators of the parameters. Two sub-models are studied using the introduced approach.