A Bayesian Inference Approach for Bivariate Weibull Distributions Derived from Roy and Morgenstern Methods

Bivariate lifetime distributions are of great importance in studies related to interdependent components, especially in engineering applications. In this paper, we introduce two bivariate lifetime assuming three- parameter Weibull marginal distributions. Some characteristics of the proposed distributions as the joint survival function, hazard rate function, cross factorial moment and stress-strength parameter are also derived. The inferences for the parameters or even functions of the parameters of the models are obtained under a Bayesian approach. An extensive numerical application using simulated data is carried out to evaluate the accuracy of the obtained estimators to illustrate the usefulness of the proposed methodology. To illustrate the usefulness of the proposed model, we also include an example with real data from which it is possible to see that the proposed model leads to good fits to the data.

Extending the survival signature paradigm to complex systems with non-repairable dependent failures

Dependent failures impose severe consequences on a complex system’s reliability and overall performance, and a realistic assessment, therefore, requires an adequate consideration of these failures. System survival signature opens up a new and efficient way to compute a system’s reliability, given its ability to segregate the structural from the probabilistic attributes of the system. Consequently, it outperforms the well-known system reliability evaluation techniques, when solicited for problems like maintenance optimisation, requiring repetitive system evaluations. The survival signature, however, is premised on the statistical independence between component failure times and, more generally, on the theory of weak exchangeability, for dependent component failures. The assumption of independence is flawed for most realistic engineering systems while the latter entails the painstaking and sometimes impossible task of deriving the joint survival function of the system components. This article, therefore, proposes a novel, generally applicable, and efficient Monte Carlo Simulation approach that allows the survival signature to be intuitively used for the reliability evaluation of systems susceptible to induced failures. Multiple component failure modes, as well, are considered, and sensitivities are analysed to identify the most critical common-cause group to the survivability of the system. Examples demonstrate the superiority of the approach.

Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components

The signature of a system is defined as the vector whose ith element is the probability that the system fails concurrently with the ith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components

The signature of a system is defined as the vector whoseith element is the probability that the system fails concurrently with theith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

AGING FUNCTIONS AND MULTIVARIATE NOTIONS OF NBU AND IFR

For d≥2, let X=(X1, …, Xd) be a vector of exchangeable continuous lifetimes with joint survival function $\overline{F}$. For such models, we study some properties of multivariate aging of $\overline{F}$ that are described by means of the multivariate aging function $B_{\overline{F}}$, which is a useful tool for describing the level curves of $\overline{F}$. Specifically, the attention is devoted to notions that generalize the univariate concepts of New Better than Used and Increasing Failure Rate. These multivariate notions are satisfied by random vectors whose components are conditionally independent and identically distributed having univariate conditional survival function that is New Better than Used (respectively, Increasing Failure Rate). Furthermore, they also have an interpretation in terms of comparisons among conditional survival functions of residual lifetimes, given a same history of observed survivals.

Bounds for the Joint Survival and Incidence Functions Through Coherent System Data

In the series system (competing risks) set-up the observed data are generally accepted as the lifetime (T) and the identifier (δ) of the component causing the failure of the system. Peterson (1976) has provided bounds for the joint survival function of the component lifetimes in terms of the joint distribution of (T, δ). In the case of more complex coherent systems, there are various schemes of observation in the literature. In this paper we provide bounds for the joint and marginal survival functions of the component lifetimes in terms of the joint distribution of the data as obtained under existing and new schemes of observation. We also tackle the reverse problem of obtaining bounds for the joint distributions of the data for given marginal distributions of the component lifetimes and the distribution of the system lifetimes.