Properties of Coherent Systems with Dependent Components

Abstract

We introduce two new families of univariate distributions that we call hyperminimal and hypermaximal distributions. These families have interesting applications in the context of reliability theory in that they contain that of coherent system lifetime distributions. For these families, we obtain distributions, bounds, and moments. We also define the minimal and maximal signatures of a coherent system with exchangeable components which allow us to represent the system distribution as generalized mixtures (i.e., mixtures with possibly negative weights) of series and parallel systems. These results can also be applied to order statistics (k-out-of-n systems). Finally, we give some applications studying coherent systems with different multivariate exponential joint distributions.

Keywords:

• Coherent systems
• Generalized mixtures
• Order statistics
• Reliability bounds

Mathematics Subject Classification:

• Primary 60E15
• Secondary 60K10

Acknowledgments

The authors wish to thank the anonymous referees for several helpful comments. Jorge Navarro and Jose M. Ruiz are partially supported by Ministerio de Ciencia y Tecnologia under grant BFM2003-02947 and Fundacion Seneca under grant 00698/PI/04.