Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T07:51:59.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Some properties of ageing notions based on the moment-generating-function order

Part of: Special processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

Xiaohu Li
Xiaohu Li*
Affiliation:
Lanzhou University
*
∗ Postal address: Department of Mathematics, Lanzhou University, Lanzhou 730000, P. R. China. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Classes of life distributions based on the moment-generating-function order are investigated in this paper. It is shown firstly that the class ℳ is closed under both convex linear combination and geometric compounding. Secondly, the class NBUmg (new better than used in the moment-generating-function order) is proved to be closed under increasing star-shaped transformations. Finally, the interplay between the stochastic comparison of the excess lifetime of a renewal process and the NBUmg interarrivals is studied.

Keywords

Convex linear combinationconvolutiongeometric compoundingℳ0-classℳ-classNBUmgstar-shaped transform

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 927 - 934
Copyright
Copyright © Applied Probability Trust 2004 

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

Alzaid, A., Kim, J. S., and Proschan, F. (1991). Laplace ordering and its applications. J. Appl. Prob. 28, 116–130.10.2307/3214745Google Scholar
Barlow, R. E., and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD.Google Scholar
Belzunce, E. M., Ortega, M., and Ruiz, J. M. (1999). Laplace order and ordering of residual lives. Statist. Prob. Lett. 42, 145–156.10.1016/S0167-7152(98)00202-8Google Scholar
Belzunce, E. M., Ortega, M., and Ruiz, J. M. (2001). A note on stochastic comparisons of excess lifetimes of renewal processes. J. Appl. Prob. 38, 747–753.10.1239/jap/1005091037Google Scholar
Bhattacharjee, M. C., Ravi, S., Vasudeva, R., and Mohan, N. R. (2003). New order preserving properties of geometric compounds. Statist. Prob. Lett. 64, 113–120.10.1016/S0167-7152(03)00039-7Google Scholar
Brown, M. (1980). Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.10.1214/aop/1176994773Google Scholar
Chen, Y. (1994). Classes of life distributions and renewal counting process. J. Appl. Prob. 31, 1110–1115.10.2307/3215334Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications. Kluwer, Dordrecht.10.1007/978-94-017-1693-2Google Scholar
Klar, B. (2002). A note on the L-class of life distributions. J. Appl. Prob. 39, 11–19.10.1239/jap/1019737984Google Scholar
Klar, B. and Müller, A. (2003). Characterizations of classes of lifetime distributions generalizing the NBUE class. J. Appl. Prob. 40, 20–32.10.1239/jap/1044476825Google Scholar
Klefsjö, B. (1982). The HNBUE and HNWUE classes of life distributions. Naval Res. Logistics Quart. 29, 331–344.10.1002/nav.3800290213Google Scholar
Klefsjö, B. (1983). A useful ageing property based on the Laplace transform. J. Appl. Prob. 20, 615–626.10.2307/3213897Google Scholar
Li, X., and Kochar, S. (2001). Some new results involving the NBU(2) class of life distributions. J. Appl. Prob. 38, 242–247.10.1239/jap/996986658Google Scholar
Li, X., Li, Z., and Jing, B. (2000). Some results about the NBUC class of life distributions. Statist. Prob. Lett. 46, 229–237.10.1016/S0167-7152(99)00104-2Google Scholar
Lin, G. D., and Hu, C.-Y. (2000). A note on the L-class of life distributions. Sankhya A 62, 267–272.Google Scholar
Müller, A., and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, New York.Google Scholar
Pellerey, F. (2000). Random vectors with HNBUE-type marginal distributions. Statist. Prob. Lett. 50, 265–271.10.1016/S0167-7152(00)00102-4Google Scholar
Rolski, T. (1975). Mean residual life. Bull. Internat. Statist. Inst. 4, 266–270.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, San Diago.Google Scholar
Shaked, M., and Zhu, H. (1992). Some results on block replacement policies and renewal theory. J. Appl. Prob. 29, 932–946.10.2307/3214725Google Scholar
Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, New York.Google Scholar
Wang, W. Y. (1996). Life distribution classes and two-unit standby redundant system. , Chinese Academy of Science, Beijing.Google Scholar
Willmot, G. E. (2002a). Compound residual lifetime distributions and the deficit at ruin. Insurance Math. Econom. 30, 421–438.10.1016/S0167-6687(02)00122-1Google Scholar
Willmot, G. E. (2002b). On higher-order properties of compound geometric distributions. J. Appl. Prob. 39, 342–340.10.1239/jap/1025131429Google Scholar
Yue, D., and Cao, J. (2001). The NBUL class of life distribution and replacement policy comparisons. Naval Res. Logistics 48, 578–591.10.1002/nav.1035Google Scholar