Hostname: page-component-89b8bd64d-b5k59 Total loading time: 0 Render date: 2026-05-07T23:17:38.539Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERIZATIONS OF THE RHR AND MIT ORDERINGS AND THE DRHR AND IMIT CLASSES OF LIFE DISTRIBUTIONS

Published online by Cambridge University Press:  31 August 2005

I. A. Ahmad  and
M. Kayid
I. A. Ahmad
Affiliation:
Department of Statistics and Actuarial Science, University of Central Florida, Orlando, Florida 32816-2370, E-mail: iahmad@mail.ucf.edu
M. Kayid
Affiliation:
Department of Mathematics, Faculty of Education (Suez), Suez Canal University, Suez, Egypt, E-mail: drkayid@yahoo.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Two well-known orders that have been introduced and studied in reliability theory are defined via stochastic comparison of inactivity time: the reversed hazard rate order and the mean inactivity time order. In this article, some characterization results of those orders are given. We prove that, under suitable conditions, the reversed hazard rate order is equivalent to the mean inactivity time order. We also provide new characterizations of the decreasing reversed hazard rate (increasing mean inactivity time) classes based on variability orderings of the inactivity time of k-out-of-n system given that the time of the (n − k + 1)st failure occurs at or sometimes before time t ≥ 0. Similar conclusions based on the inactivity time of the component that fails first are presented as well. Finally, some useful inequalities and relations for weighted distributions related to reversed hazard rate (mean inactivity time) functions are obtained.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 19 , Issue 4 , October 2005 , pp. 447 - 461
Copyright
© 2005 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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Ahmad, I.A., Kayid, M., & Pellerey, F. (2005). Further results involving the MIT order and the IMIT class. Probability in the Engineering and Informational Sciences 19: 377–395.Google Scholar
Alzaid, A. (1988). Mean residual life ordering. Statistical Papers 25: 477–482.Google Scholar
Alzaid, A., Kim, J.S., & Proschan, F. (1991). Laplace ordering and its applications. Journal of Applied Probability 28: 116–130.Google Scholar
Andersen, P.K., Borgan, O., Gill, R.D., & Keiding, N. (1993). Statistical models based on counting processes. New York: Springer Verlag.CrossRef
Asadi, M. (2005). On the mean past lifetime of the components of a parallel system. Journal of Statistical Planning and Inference, to appear.Google Scholar
Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing. Silver Spring, MD: To Begin with.
Bartoszewicz, J. & Skolimowska, M. (2005). Preservation of classes of life distributions under weighting. Statistics & Probability Letters, to appear.Google Scholar
Belzunce, F., Franco, M., & Ruiz, J.M. (1999). On aging properties based on the residual life of k-out-of-n systems. Probability in the Engineering and Informational Sciences 13: 193–199.Google Scholar
Belzunce, F., Gao, X., Hu, T., & Pellerey, F. (2004). Characterizations of the hazard rate and IFR aging notion. Statistics & Probability Letters 40: 235–242.Google Scholar
Belzunce, F., Navarro, J., Ruiz, J.M., & del Aguila, Y. (2004). Some results on residual entropy function. Metrika 59: 147–161.Google Scholar
Block, H.W., Savits, T.H., & Singh, H. (1998). The reversed hazard rate function. Probability in the Engineering and Informational Sciences 12: 69–70.Google Scholar
Broderick, O. & Olusegun, G. (2002). On stochastic inequalities and comparisons of reliability measures for weighted distributions. Mathematical Problems in Engineering 8: 1–13.Google Scholar
Chandra, N.K. & Roy, D. (2001). Some results on reversed hazard rate. Probability in the Engineering and Informational Sciences 15: 95–102.Google Scholar
Denuit, M. (2001). Laplace transform ordering of actuarial quantities. Insurance: Mathematics and Economics 29: 83–102.Google Scholar
Di Crescenzo, A. & Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distribution. Journal of Applied Probability 39: 434–440.Google Scholar
Fernandez-Ponce, J.M., Kochar, S.C., & Munoz-Pérez, J. (1998). Partial orderings of distributions based on right spread functions. Journal of Applied Probability 35: 221–228.Google Scholar
Jain, K., Singh, H., & Bagai, I. (1989). Relations for reliability measures of weighted distributions. Communications in Statistics—Theory and Methods 18: 4393–4412.Google Scholar
Kalbfleisch, J.D. & Lawless, J.F. (1989). Inference based on retrospective ascertainment: An analysis of the data on transfusion-related AIDS. Journal of the American Statistical Association 84: 360–372.Google Scholar
Kayid, M. & Ahmad, I.A. (2004). On the mean inactivity time ordering with reliability applications. Probability in the Engineering and Informational Sciences 18: 395–409.Google Scholar
Klefsjo, B. (1983). A useful ageing property based on the Laplace transform. Journal of Applied Probability 20: 615–626.Google Scholar
Kochar, S.C., Li, X., & Shaked, M. (2002). The total time on test transform and the excess wealth stochastic orders of distributions. Advances in Applied Probability 34: 826–845.Google Scholar
Li, X. & Chen, J. (2004). Aging properties of the residual life length of k-out-of-n systems with independent but non-identical components. Applied Stochastic Models in Business and Industry 20: 143–153.Google Scholar
Li, X. & Lu, J. (2003). Stochastic comparisons on residual life and inactivity time of series and parallel systems. Probability in the Engineering and Informational Sciences 17: 267–275.Google Scholar
Li, X. & Yam, R.C.M. (2005). Reversed preservation properties of some negative aging conceptions. Statistical Papers 46: 65–78.Google Scholar
Li, X. & Zuo, M.J. (2004). Preservation of stochastic orders for random minima and maxima, with applications. Naval Research Logistics 51: 332–344.Google Scholar
Müller, A. & Stoyan, D. (2002). Comparison methods for queues and other stochastic models. New York: Wiley.
Nanda, A.K. & Jain, K. (1999). Some weighted distribution results on univariate and bivariate cases. Journal of Statistical Planning and Inference 77: 169–180.Google Scholar
Nanda, A.K., Singh, H., Misra, N., & Paul, P. (2003). Reliability properties of reversed residual lifetime. Communications in Statistics—Theory and Methods 32: 2031–2042.Google Scholar
Navarro, J., del Aguila, Y., & Ruiz, J.M. (2001). Characterizations through reliability measures from weighted distributions. Statistical Papers 42: 395–402.Google Scholar
Pakes, A.G., Sapatinas, T., & Fosam, E.B. (1996). Characterizations, length-biasing and infinite divisibility. Statistical Papers 37: 53–69.Google Scholar
Patil, G.P. & Rao, C.R. (1977). The weighted distributions: A survey and their applications. In: P.R. Krishnaiah (ed.), Applications of statistics. Amsterdam: North-Holland, pp. 385–405.
Patil, G.P. & Rao, C.R. (1978). Weighted distributions and size-biased sampling with applications to wild-life populations and human families. Biometrics 34: 179–189.Google Scholar
Pellerey, F. & Petakos, K. (2002). On closure property of the NBUC class under formation of parallel systems. IEEE Transactions on Reliability 51: 452–454.Google Scholar
Sengupta, D. & Nanda, A.K. (1999). Log-concave and concave distributions in reliability. Naval Research Logistics 46: 419–433.Google Scholar
Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. New York: Academic Press.
Xu, M. & Li, X. (2005). Behavior of negative aging properties based upon the inactivity time and residual life. Technical report at the Department of Mathematics, Lanzhou University, Lanzhou, Peoples' Republic of China.