Bivariate distribution models using copulas for reliability analysis

The modeling of joint probability distributions of correlated variables and the evaluation of reliability under incomplete probability information remain a challenge that has not been studied extensively. This article aims to investigate the effect of copulas for modeling dependence structures between variables on reliability under incomplete probability information. First, a copula-based method is proposed to model the joint probability distributions of multiple correlated variables with given marginal distributions and correlation coefficients. Second, a reliability problem is formulated and a direct integration method for calculating probability of failure is presented. Finally, the reliability is investigated to demonstrate the effect of copulas on reliability. The joint probability distribution of multiple variables, with given marginal distributions and correlation coefficients, can be constructed using copulas in a general and flexible way. The probabilities of failure produced by different copulas can differ considerably. Such a difference increases with decreasing probability of failure. The reliability index defined by the mean and standard deviation of a performance function cannot capture the difference in the probabilities of failure produced by different copulas. In addition, the Gaussian copula, often adopted out of expedience without proper validation, produces only one of the various possible solutions of the probability of failure and such a probability of failure may be biased towards the non-conservative side. The tail dependence of copulas has a significant influence on reliability.