System components usually attain marginal lifetimes with stochastic dependence in the context of load-sharing reliability structures. This study deals with the load-sharing parallel systems of two components. We prove that two marginal lifetimes are positively quadrant dependent when component lifetimes have continuous probability distributions, and such a stochastic dependence is upgraded to the total positive of order 2 in the setting of component lifetimes having an exponential distribution. In addition, we discuss how these findings shed light on related results for the load-sharing Ross model, the conditional residual lifetime, and the conditional inactivity time.

The generalized Gompertz distribution—an extension of the standard Gompertz distribution as well as the exponential distribution and the generalized exponential distribution—offers more flexibility in modeling survival or failure times as it introduces an additional parameter, which can account for different shapes of hazard functions. This enhances its applicability in various fields such as actuarial science, reliability engineering and survival analysis, where more complex survival models are needed to accurately capture the underlying processes. The effect of heterogeneity has generated increased interest in recent times. In this article, multivariate chain majorization methods are exploited to develop stochastic ordering results for extreme-order statistics arising from independent heterogeneous generalized Gompertz random variables with increased degree of heterogeneity.

In this work, by considering coherent systems comprising independent components with discrete lifetimes, we introduce the notion of discrete-time signature and then discuss some of its properties. With the use of the introduced signature, a stochastic ordering result is also established. We then introduce transformation formulas for the discrete-time signature to facilitate the comparison of systems of different sizes. Some examples are also presented to illustrate all the results developed here.

Lorenz dominance is a classical criterion for comparing income distributions with respect to inequality and social welfare. However, its binary nature, in which one distribution either dominates another or does not, often leads to inconclusive results when empirical Lorenz curves intersect. To overcome this limitation, we introduce the Lorenz dominance index (LDI), a continuous measure that quantifies the extent to which one Lorenz curve lies above another. The LDI provides an interpretable assessment based on the population, allowing for the evaluation of partial or near dominance and improving its usefulness in empirical settings. We derive the asymptotic distribution of the LDI and propose a nonparametric bootstrap procedure to construct confidence intervals and perform inference. Monte Carlo simulations confirm the estimator’s strong performance in finite samples and its nominal coverage. An application to household income data from China highlights the practical value of the LDI in distributional analysis.

It is well known that any higher-order Markov chain can be associated with a first-order Markov chain. In this primarily expository article, we present the first fairly comprehensive analysis of the relationship between higher-order and first-order Markov chains, together with illustrative examples. Our main objective is to address the central question as posed in the title.

The $q$-Weibull distribution, as a generalization of the Weibull distribution, plays an important role in the field of reliability theory, survival analysis, finance, engineering, medical science, etc. In contrast to the Weibull distribution, which is limited to describing monotonic hazard rate functions, the $q$-Weibull distribution offers the flexibility to model various behaviors of the hazard rate function, including unimodal, bathtub-shaped, monotonic (both increasing and decreasing), and constant. In this article, we investigated the stochastic comparison of extreme order statistics derived from independent, heterogeneous $q$-Weibull random variables using various stochastic orderings, including the usual stochastic order, hazard rate order, reversed hazard rate order, and likelihood ratio order. Some of these results are further extended to dependent setups by incorporating Archimedean copulas to model the dependence structure. Finally, we explored the behavior of extreme order statistics when the random variables are subjected to random shocks.

This paper investigates the complexity of residual lifetimes of live components in coherent systems through the lens of cumulative residual extropy and its divergence-based extension, Jensen-cumulative residual extropy. Unlike classical reliability metrics that focus on system inactivity or mean residual life, our framework quantifies the hidden informational structure of components that remain alive at the system failure time. We derive closed-form expressions for the cumulative residual extropy of conditional residual lifetimes using system signatures and establish stochastic bounds and comparisons that highlight the impact of structural configuration. A novel divergence measure, the Jensen-cumulative residual extropy, is introduced to capture discrepancies between coherent systems and benchmark $k$-out-of-$n$ structures. Numerical illustrations with gamma-distributed lifetimes demonstrate the sensitivity of cumulative residual extropy and Jensen-cumulative residual extropy to redundancy patterns and dependence structures. Furthermore, by integrating cost considerations into the divergence framework, we provide a rigorous optimization scheme for selecting system signatures that jointly minimize informational complexity and economic expenditure. The proposed approach enriches the theoretical foundation of reliability analysis and offers practical guidelines for designing resilient, cost-effective, and information-efficient engineering systems.

Consecutive $k$-type systems have become important in both reliability theory and applications; in spite of a large literature existing on them, three-dimensional consecutive $k$-type systems have not yet been studied for multi-state case. In this paper, we introduce several different types of multi-state linear three-dimensional consecutive $k$-type systems for the first time, with due consideration to possible overlapping of failure blocks. The finite Markov chain imbedding approach is then used for the derivation of their reliability functions with state spaces and transition matrices provided in a novel way, and the involved computational process is illustrated through several numerical examples. Finally, some possible applications of the work and potential extensions are pointed out.