The cumulative residual extropy has been proposed recently as an alternative measure of extropy to the cumulative distribution function of a random variable. In this paper, the concept of cumulative residual extropy has been extended to cumulative residual extropy inaccuracy (CREI) and dynamic cumulative residual extropy inaccuracy (DCREI). Some lower and upper bounds for these measures are provided. A characterization problem for the DCREI measure under the proportional hazard rate model is studied. Nonparametric estimators for CREI and DCREI measures based on kernel and empirical methods are suggested. Also, a simulation study is presented to evaluate the performance of the suggested measures. Simulation results show that the kernel-based estimator performs better than the empirical-based estimator. Finally, applications of the DCREI measure for model selection are provided using two real data sets.

In this paper, we use an information theoretic approach called cumulative residual extropy (CRJ) to compare mixed used systems. We establish mixture representations for the CRJ of mixed used systems and then explore the measure and comparison results among these systems. We compare the mixed used systems based on stochastic orders and stochastically ordered conditional coefficients vectors. Additionally, we derive bounds for the CRJ of mixed used systems with independent and identically distributed components. We also propose the Jensen-cumulative residual extropy (JCRJ) divergence to calculate the complexity of systems. To demonstrate the utility of these results, we calculate and compare the CRJ and JCRJ divergence of mixed used systems in the Exponential model. Furthermore, we determine the optimal system configuration based on signature under a criterion function derived from JCRJ in the exponential model.

Recently, an alternative measure of uncertainty called extropy is proposed by Lad et al. [12]. The extropy is a dual of entropy which has been considered by researchers. In this article, we introduce an alternative measure of uncertainty of random variable which we call it cumulative residual extropy. This measure is based on the cumulative distribution function F. Some properties of the proposed measure, such as its estimation and applications, are studied. Finally, some numerical examples for illustrating the theory are included.