Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T08:10:31.578Z Has data issue: false hasContentIssue false
  • English
  • Français

ROLE OF EQUILIBRIUM DISTRIBUTION IN RELIABILITY STUDIES

Published online by Cambridge University Press:  27 February 2007

Ramesh C. Gupta
Ramesh C. Gupta
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, ME 04469-5752, E-mail: ramesh_gupta@umit.maine.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The equilibrium distribution arises as the limiting distribution of the forward recurrence time in a renewal process. The purpose of this article is to study the relationships between the equilibrium distributions (including its higher derivates) and the original distributions. Some stochastic order relations and the relations between their aging properties are investigated and some applications in the field of insurance and financial investments are given. In addition, the relation between the equilibrium distribution of a series system and the series system of equilibrium distribution is investigated. Bivariate equilibrium distribution whose reliability properties are consistent with those of the univariate equilibrium distribution is defined.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 21 , Issue 2 , April 2007 , pp. 315 - 334
Copyright
© 2007 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abouammoh, A.M. (1988). On the criteria of the mean remaining life. Statistics and Probability Letters 6: 205–211.Google Scholar
Abouammoh, A.M. & Ahmed, A.N. (1988). The new better than used failure rate class of life distributions. Advances in Applied Probability 20: 237–240.Google Scholar
Abouammoh, A.M., Kanjo, A., & Khalique, A. (1990). On aspects of variance remaining life distributions. Microelectron Reliability 30(4): 751–776.Google Scholar
Arnold, B.C. (1995). Conditional survival models. In N. Balakrisnan (ed.), Recent advances in life testing and reliability. Boca Raton, FL: CRC Press, pp. 589–601.
Basu, A.P. & Ebrahimi, N. (1984). On k-order harmonic new better than used in expectation distribution. Annals of the Institute of Statistical Mathematics A 36: 87–100.Google Scholar
Basu, A.P. & Ebrahimi, N. (1985). Corrections to “on k-order harmonic new better than used in expectation distributions.” Annals of the Institute of Statistical Mathematics A 37: 365–366.Google Scholar
Belzunce, F., Candel, J., & Ruiz, J.M. (1996). Dispersive orderings and characterizations of ageing classes. Statistics and Probability Letters 28: 321–327.Google Scholar
Bhattacharjee, M.C. & Sethuraman, J. (1990). Families of life distributions characterized by two moments. Journal of Applied Probability 27: 720–725.Google Scholar
Bon, J-L. & Illayk, A. (2005). Ageing properties and series systems. Journal of Applied Probability 42: 279–286.Google Scholar
Brown, M. (2006). Exploiting the waiting time paradox. Probability in the Engineering and Informational Sciences 20: 195–230.Google Scholar
Bryson, M.C. & Siddiqi, M.M. (1969). Some criteria for aging. Journal of the American Statistical Association 64: 1472–1483.Google Scholar
Deshpande, J.V., Kochar, S.C., & Singh, H. (1986). Aspects in positive ageing. Journal of Applied Probability 23: 74–78.Google Scholar
Deshpande, J.V., Singh, H., Bagai, I., & Jain, K. (1990). Some partial orderings describing positive ageing. Communications in Statistics: Stochastic Models 6(3): 471–481.Google Scholar
Fagiuoli, E. & Pellerey, F. (1993). New partial orderings and applications. Naval Research Logistics Quarterly 40: 829–842.Google Scholar
Gupta, P.L. & Gupta, R.C. (1983). On the moments of residual life in reliability and some characterization results. Communications in Statistics: Theory and Methods 12: 449–461.Google Scholar
Gupta, R.C. (1987). On the monotonic properties of residual variance and their applications in reliability. Journal of Statistical Planning and Inference 16: 329–335.Google Scholar
Gupta, R.C. & Kirmani, S.N.U.A. (1998). Residual life function in reliability studies. In A.P. Basu, S.K. Basu, & S. Mukhopadhyay (eds.), Frontiers in reliability. River Edge, NJ: World Scientific.
Gupta, R.C. & Kirmani, S.N.U.A. (2000). Residual coefficient of variation and some characterization results. Journal of Statistical Planning and Inference 91: 23–31.Google Scholar
Gupta, R.C., Kirmani, S.N.U.A., & Launer, R.L. (1987). On life distributions having monotone residual variance. Probability in the Engineering and Informational Sciences 1: 299–307.Google Scholar
Gupta, R.C., Tajdari, M., & Bresinsky, H. (2006). Some general results for moments in bivariate distributions. Submitted for publication.
Gupta, R.P. & Sankaran, P.G. (1998). Bivariate equilibrium distribution and its applications to reliability. Communications in Statistics: Theory and Methods 27(2): 385–394.Google Scholar
Hesselager, O. (1995). Order relations for some distributions. Insurance: Mathematics and Economics 16: 129–134.Google Scholar
Hesselager, O., Wang, S., & Willmot, G. (1997). Exponential and scale mixtures and equilibrium distributions. Scandinavian Acturial Journal 2: 125–142.Google Scholar
Hutchinson, T.P. & Lee, C.D. (1991). The engineering statistician's guide to continuous bivariate distributions. Adelaide, Australia: Rumsby Scientific Publishing.
Klar, B. & Muller, A. (2003). Characterizations of classes of lifetime distributions generalizing the NBUE class. Journal of Applied Probability 40: 20–32.Google Scholar
Klefsjo, B. (1982). The HNBUE and HNWUE classes of distributions. Naval Research Logistics Quarterly 29: 331–344.Google Scholar
Klefsjo, B. (1983). A useful ageing property based on Laplace transform. Journal of Applied Probability 20: 615–626.Google Scholar
Launer, R.L. (1984). Inequalities for NBUE and NWUE life distributions. Operations Research 32: 660–667.Google Scholar
Li, H. & Zhu, H. (1994). Stochastic equivalence of ordered random variables with applications in reliability theory. Statistics and Probability Letters 20: 383–393.Google Scholar
Loh, W.Y. (1984). A new generalization of the class of NBU distributions. IEEE Transactions on Reliability 33: 419–422.Google Scholar
Navarro, J., Franco, M., & Ruiz, J.M. (1998). Characterization through moments of the residual life and conditional spacings. Sankhya, Series A 60: 36–48.Google Scholar
Shaked, M. (1981). Extensions of IHR and IHRA ageing notions. SIAM Journal of Applied Mathematics 40(3): 542–550.Google Scholar
Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. New York: Academic Press.
Singh, H. (1989). On partial ordering of life distributions. Naval Research Logistics Quarterly 36: 103–110.Google Scholar
Singh, H. & Deshpande, J.V. (1985). On some new ageing properties. Scandinavian Journal of Statistics 12: 213–220.Google Scholar
Stein, W.E. & Dattero, R. (1999). Bondesson's functions in reliability theory. Applied Stochastic Models in Business and Industry 15: 103–109.Google Scholar
Willmot, G.E. (1997). Bounds for compound distributions based on mean residual lifetimes and equilibrium distributions. Insurance: Mathematics and Economics 21: 25–42.Google Scholar
Willmot, G.E., Drekic, S., & Cai, J. (2005). Equilibrium compound distributions and stop-loss moments. Scandinavian Acturial Journal 1: 6–24.Google Scholar
Zarek, M. (1995). Some generalized variability orderings among life distributions with applications to Weibull's and gamma distribution. Fasciculi Mathematics 25: 197–209.Google Scholar