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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.

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In this paper, a novel index entitled the Jensen cumulative residual extropy divergence is investigated for the analysis and measurement of the behavioral complexity of conditional mixed systems. First, using the vector of conditional coefficients obtained from the signature vector, the behavior of this measure is analytically examined for a class of coherent systems as well as their dual systems, in the case where the components follow gamma distributions. Then, simulations are performed to evaluate the obtained results. The results of this paper show that the minimum complexity is achieved by coherent $k$-out-of-$n$ systems with the Jensen cumulative residual extropy divergence equal to zero. Moreover, the results indicate that duality of systems does not necessarily lead to equality of the Jensen cumulative residual extropy divergence in conditional mixed systems; rather, this index is sensitive to component weighting, order statistics, and the structural interaction among the components of the system.