Abstract
The life lengths of some possibly dependent components in a system can be modelled by a multivariate distribution. In this paper, we suppose that the joint distribution of the units is a symmetric multivariate Gumbel exponential distribution (GED). Hence, the components are exchangeable and have exponential (marginal) distributions. For this model, we obtain basic reliability properties for k-out-of-n systems (order statistics) and, in particular, for series and parallel systems. We pay special attention to systems with two components. Some results are extended to coherent systems with n exchangeable components.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ashour, S. K., and Youssef, A. (1991). Bayesian estimation of a linear failure rate, Journal of the Indian Association for Production, Quality and Reliability, 16, 9–16.
Baggs, G. E., and Nagaraja, H. N. (1996). Reliability properties of order statistics from bivariate exponential distributions, Communications in Statistics — Stochastic Models, 12, 611–631.
Bain, L. J. (1974). Analysis for the linear failure-rate life-testing distribution, Technometrics, 16, 551–559.
Balakrishnan, N., Bendre, S. M., and Malik, H. J. (1992). General relations and identities for order statistics from non-independent non-identical variables, Annals of the Institute of Statistical Mathematics, 44, 177–183.
Barlow, R. E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, New York.
Belzunce, F. Franco, M., Ruiz, J. M., and Ruiz, M. C. (2001). On partial orderings between coherent systems with different structures, Probability in the Engineering and Informational Sciences, 15, 273–293.
Block, H. W., and Joe, H. (1997). Tail behavior of the failure rate functions of mixtures, Lifetime Data Analysis, 3, 269–288.
Castillo, E., Sarabia, J. M., and Hadi, A. S. (1997). Fitting continuous bivariate distributions to data. The Statistician, 46, 355–369.
Cox, D. R., and Oakes, D. (1984). Analysis of Survival Data, Chapman and Hall, London.
David, H. A. (1970). Order Statistics, John Wiley& Sons, New York.
Gumbel, E. J. (1960). Bivariate exponential distributions, Journal of the American Statistical Association, 55, 698–707.
Gupta, R. C. (2001). Reliability studies of bivariate distributions with Pareto conditionals, Journal of Multivariate Analysis, 76, 214–225.
Gupta, P. L., and Gupta, R. C. (2001). Failure rate of the minimum and maximum of a multivariate normal distribution, Metrika, 53, 39–49.
Kanjo, A. I., and Abouammoh, A. M. (1995). Closure of mean remaining life classes under formation of parallel systems, Pakistan Journal of Statistics, 11(2), 153–158.
Kochar, S., Mukerjee, H., and Samaniego, F. J. (1999). The “signature” of a coherent system and its application to comparison among systems, Naval Research Logistics, 46, 507–523.
Kotz, S., Balakrishnan, N., and Johnson, N. L. (2000). Continuous Multivariate Distributions, John Wiley_& Sons, New York.
Kotz, S., Lai, C. D. and Xie, M. (2003). On the effect of redundancy for systems with dependent components, IIE Transactions, 35, 1103–1110.
Maurer, W., and Margolin, B. H. (1976). The multivariate inclusion-exclusion formula and order statistics from dependent variates, Annals of Statistics, 4, 1190–1199.
Mi, J., and Shaked, M. (2002). Stochastic dominance of random variables implies the dominance of their order statistics, Journal of the Indian Statistical Association, 40, 161–168.
Roy, D. (2001). Some properties of a classification system for multivariate life distributions, IEEE Transactions on Reliability, 50, 214–220.
Rychlik, T. (2001a). Stability of order statistics under dependence, Annals of the Institute of Statistical Mathematics, 53, 877–894.
Rychlik, T. (2001b). Mean-variance bounds for order statistics from dependent DFR, IFR, DFRA and IFRA samples, Journal of Statistical Planning and Inference, 92, 21–38.
Samaniego, F. (1985). On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability, 34, 69–72.
Sen, A., and Bhattacharyya, G. K. (1995). Inference procedures for the linear failure rate model, Journal of Statistical Planning and Inference, 46, 59–76.
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications, Academic Press, San Diego.
Shaked, M., and Suarez-Llorens, A. (2003). On the comparison of reliability experiments based on the convolution order. Journal of the American Statistical Association, 98, 693–702.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Navarro, J., Ruiz, J.M., Sandoval, C.J. (2006). Systems with Exchangeable Components and Gumbel Exponential Distribution. In: Balakrishnan, N., Sarabia, J.M., Castillo, E. (eds) Advances in Distribution Theory, Order Statistics, and Inference. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4487-3_19
Download citation
DOI: https://doi.org/10.1007/0-8176-4487-3_19
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4361-4
Online ISBN: 978-0-8176-4487-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)