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.

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.

In this paper, quantile-based dynamic cumulative residual and failure extropy measures are introduced. For a presentation of their applications, first, by using the simulation technique, a suitable estimator is selected to estimate these measures from among different estimators. Then, based on the equality of two extropy measures in terms of order statistics, symmetric continuous distributions are characterized. In this regard, a measure of deviation from symmetry is introduced and how it is applied is expressed in a real example. Also, among the common continuous distributions, the generalized Pareto distribution and as a result the exponential distribution are characterized, and based on the obtained results, the exponentiality criterion of a distribution is proposed.

This paper introduces the dynamic weighted cumulative residual extropy criterion as a generalization of the weighted cumulative residual extropy criterion. The relationship of the proposed criterion with reliability criteria such as weighted mean residual lifetime, hazard rate function, and second-order conditional moment are studied. Also, characterization properties, upper and lower bounds, inequalities, and stochastic orders based on dynamic weighted cumulative residual extropy and the effect of linear transformation on it will be presented. Then, a non-parametric estimator based on the empirical method for the introduced criterion is given, and its asymptotic properties are studied. Finally, an application of the dynamic weighted cumulative residual extropy in selecting the appropriate data distribution on a real data set is discussed.

AbstractThe aim of this study was to investigate the validity, reliability and factor structure of the "How I Think" questionnaire to assess cognitive distortions in Iranian students. The present research design is descriptive and psychometric studies. The statistical population of this study included all male high school students in city Dezful in the academic year 1399-1400, from which 316 students were selected by multi-stage random sampling and completed the How I think and Bass and Perry Aggression Questionnaires (for Convergent validity). Factor analysis, Cronbach's alpha and Pearson correlation were used to analyze the data using SPSS and AMOS 20 software. The results of factor analysis showed that the questionnaire consists of six factors and has good fit indices. The convergent validity of this questionnaire indicated a satisfactory correlation between the dimensions of the scale "How I think" with the Aggression Questionnaire (p <0.01). Cronbach's alpha and Split- half coefficients for the whole scale and its dimensions ranged from 0.66 to 0.92.The conclusion is that the Persian version of the "How do I think" questionnaire for measuring cognitive distortions in Iranian students has acceptable psychometric properties and can be used as a valid tool in research, educational and clinical situations.

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.

In this paper, we first of all define the distance measure entitled generalized Hausdorff distance between two trapezoidal generalized fuzzy numbers (TGFNs) that has been introduced by Chen [10]. Then using a other distance and combining with generalized Hausdorff distance, we define the similarity measure. The basic properties of the above mentioned similarity measure are proved in detail. Finally we rank two generalized fuzzy numbers using distance measure and similarity measure between them.

For a given measure space (X, B, m) we construct all measure spaces (Y, C, l) in which (X, B, m) is embeddable. The construction is modeled on the ultrafilter construction of the Stone-Cech compactification of a completely regular topological space.Under certain conditions the construction simplifies. Examples are given when this simplification occurs.