Compounded generalized Weibull distributions-A unified approach

In this paper we analyze a unified approach to study a family of lifetime distributions of a system consisting of random number of components in series and in parallel proposed by Chowdhury (2014). While the lifetimes of the components are assumed to follow generalized (exponentiated) Weibull distribution, a zero-truncated Poisson is assigned to model the random number of components in the system. The resulting family of compounded distributions describes several well-known distributions as well as some new models. Bivariate extension of the proposed family of distribution is also introduced. Some important statistical and reliability properties of the family of distributions are investigated. The distribution is found to exhibit both monotone and non-monotone failure rates. Parameters of the proposed distributions are estimated by the expectation maximization (EM) algorithm. Some numerical results are obtained through Monte-Carlo simulation. The asymptotic variance-covariance matrices of the estimators are also derived. Potential of the distribution is explored through two real data sets and compared with similar compounded distribution and the results demonstrate that the family of distributions can be considered as a suitable model under several real situations.