In this paper, we characterize several partial dependencies in a general mixture model of weighted distributions with a parametric weight function that encompasses many well-known frailty models. There are well-known frailty models in survival analysis satisfying the proposed mixture model which are used to examine the results. The mixture-based copula functions associated with the mixture model are characterized. Examples are given to draw the copula functions out from respected mixture models.

Statistical reliability modeling of lifetime data routinely encounters distributions that are asymmetric on the original scale yet exhibit fundamental structural balance after logarithmic transformation. Addressing this intrinsic feature of reliability and survival data, we developed a unified entropy-based reliability modeling framework for the characterization and testing of log-symmetry in continuous lifetime distributions. The proposed methodology was built upon distributional transformations induced by linear consecutive k-out-of-n reliability systems, which serve as structured mechanisms for probing how informational balance is preserved or disrupted under reliability-driven system behavior. Within this framework, Shannon entropy, Rényi entropy, and Kerridge inaccuracy were integrated to derive explicit and tractable characterization results that uniquely identified log-symmetric lifetime distributions through intrinsic information-theoretic relationships. These results led naturally to a nonparametric, computationally efficient statistical test for log-symmetry that avoided restrictive modeling assumptions and was well suited for practical reliability analysis. Comprehensive Monte Carlo simulations demonstrated that the proposed test achieved accurate type I error control and competitive power against a wide range of alternatives. Applications to real lifetime datasets from reliability settings further confirmed the effectiveness of the approach. Overall, the study demonstrated how entropy-based inference, when embedded within reliability system modeling, provides a rigorous and practically relevant framework for statistical analysis of lifetime distributions, thereby contributing theoretical insight and applied methodology to modern reliability modeling.

In this paper, a mean inactivity time frailty model is considered. Examples are given to calculate the mean inactivity time for several reputable survival models. The dependence structure between the population variable and the frailty variable is characterized. The classical weighted proportional mean inactivity time model is considered as a special case. We prove that several well-known stochastic orderings between two frailties are preserved for the response variables under the weighted proportional mean inactivity time model. We apply this model on a real data set and also perform a simulation study to examine the accuracy of the model.

This study explores the extropy of consecutive r-out-of-n:G systems, offering a detailed framework for theoretical analysis and practical applications. Exact expressions for system lifetime extropy are derived, with comparative evaluations across diverse lifetime distributions. Theoretical contributions include bounds, characterization results, and insights into the variability of extropy. Practically, a nonparametric extropy estimator is introduced and validated through simulations and image processing applications. A novel test statistic for exponentiality is also proposed, with the critical values computed numerically and the performance assessed against alternative distributions. The results highlight the test's superior efficacy in specific contexts while noting its limitations. This work combines theoretical and practical advances, providing valuable tools for reliability analysis and statistical inference.

In most lifetime models, the bivariate $\alpha$-quantile inactivity time is a vector of increasing functions. A novel estimator of this vector was created and investigated under this assumption. It was expected that the application of this knowledge would improve the efficiency of the estimator. It was proven that the proposed estimator is consistent and converges weakly to a bivariate Gaussian process under a suitable transformation. A simulation study was conducted to compare the performance of the proposed estimator with that of the usual estimator. Finally, the application of the proposed estimator is illustrated by analyzing a dataset comprising the time to blindness in patients with diabetic retinopathy.

In this study, we focused on investigating the properties of residual Tsallis entropy for order statistics. The reliability of engineering systems is highly influenced by order statistics, for example, when modeling the lifetime of a series system and the lifetime of a parallel system. The residual Tsallis entropy of the ith order statistic from a continuous distribution function and its deviation from the residual Tsallis entropy of the ith order statistics from a uniform distribution were investigated. In the mathematical framework, a method was provided to represent the residual Tsallis entropy of the ith order statistic in the continuous case with respect to the case where the distribution was uniform. This approach can provide insight into the behavior and properties of the residual Tsallis entropy for order statistics. We also investigated the monotonicity of the new uncertainty measure under different conditions. An investigation of these properties leads to a deeper understanding of the relationship between the position of the order statistics and the resulting Tsallis entropy. Finally, we presented the computational results and proposed estimators for estimating the residual Tsallis entropy of an exponential distribution. For this purpose, we derived a maximum likelihood estimator.

Recently, extropy has emerged as an alternative measure of uncertainty instead of entropy. When it comes to quantifying uncertainty regarding the remaining lifetime of a component, entropy has proven to be less effective. Therefore, the concept of residual entropy was introduced to address this limitation. Similar to the residual entropy, the residual extropy was formulated and used to investigate the uncertainty in the residual lifetime of a unit. Systems in the real world exhibit a pervasive property of uncertainty that affects future events and past events. For this reason, the concept of past extropy was introduced to specifically capture and analyze the uncertainty associated with past events. This paper focuses on stochastic aspects, including stochastic orderings, which provide useful inequalities related to past extropy when applied to order statistics and lower record values. It is worth noting that the past extropy of the $i$th-order statistics and record values in the continuous case is related to the past extropy of the $i$th-order statistics and record values evaluated from the uniform distribution. The monotonicity of the past extropy of order statistics is examined and some insights into the past extropy of lower data set values are also given. Finally, some computational results are presented. In fact, an estimator for the extropy of the exponential distribution is proposed. For this purpose, the maximum likelihood estimator is derived. The proposed method is easy to implement and apply from a computational point of view.

In reliability engineering and survival analysis, quantile functions are fundamental and often the most natural way to represent probability distributions and data samples. In this paper, the $\alpha$-quantile function of past lifetime was estimated for right-censored data by applying the Kaplan-Meier survival estimator. The weak convergence of the proposed estimator to a Gaussian process was investigated. A confidence interval for the $\alpha$-quantile of the past life function that does not depend on the density function was proposed. The strong convergence of the estimator to a Gaussian process was also discussed. The properties of the estimator and the confidence interval were investigated in a simulation study. Finally, two real datasets were analyzed.

To investigate potentially dependent lifetimes, it is necessary to extend the $\alpha$-quantile inactivity time to bivariate and multivariate frameworks. To extend this measure to a dynamic multivariate framework, all possible trajectories at time $t$ are considered. The behavior of the extended $\alpha$-quantile of inactivity time was investigated in relation to the corresponding multivariate hazard rate function. The $\alpha$-quantile of the inactivity order is defined and discussed for the multivariate case. The difference between the two bivariate $\alpha$-quantile functions of inactivity, which is an important measure for studying the effect of treatment on lifespan, was also investigated. This measure was used to analyze the effect of laser treatment on the delay of blindness. Two bootstrap approaches were implemented to construct confidence bounds for the difference measure.