In this paper, we further investigate stochastic comparisons of the lifetime of parallel systems with heterogeneous independent Pareto components in term of the star order and convex order. It will be proved that the lifetime of a parallel system with heterogeneous independent components from Pareto model is always smaller than from the lifetime of another parallel system with homogeneous independent components from Pareto model in the sense of convex order. Also, under a general condition on the scale parameters, it is proved a result involving with star order.

This paper deals with some stochastic comparisons of convolution of random variables comprising scale variables. Sufficient conditions are established for these convolutions' likelihood ratio ordering and hazard rate order. The results established in this paper generalize some known results in the literature. Several examples are also presented for more illustrations.

In this paper, a nonparametric test based on incomplete data is proposed to investigate the usual stochastic order  using an extension of Banerjee statistic for Type I censored data. This extension is optimized with weight coefficients based on Simpson's rule and the bootstrap method with 10000 iterations to estimate the empirical distribution of the proposed test statistic. The empirical distribution of the statistic under censoring is studied, and the power of the test is evaluated using Monte Carlo simulations against the Lehmann alternative model.

‎Large engineering infrastructures such as power grids‎, ‎communication systems‎, ‎and cyber-physical systems are exposed to shocks of varying intensity and scope‎, ‎which due to the dependence between components‎, ‎can cause successive failures throughout the entire network‎. ‎The classical reliability models are generally based on the assumption of independence between components and shocks and therefore face limitations in describing the real behavior of the systems‎. ‎In this paper‎, ‎a comprehensive framework for analyzing the reliability of networks in the presence of signed shocks time-dependent is presented‎. ‎In this framework‎, ‎each network node is modeled by a signed process that simultaneously captures the magnitude and timing of shocks‎, ‎and the dependence structure between components is described through a copula function‎. ‎In addition to statistical dependence‎, ‎the proposed model also includes cumulative failure mechanisms‎, ‎incomplete repair‎, ‎and reinforcement resource allocation policies.