Hostname: page-component-7b7567c8c-lpg2g Total loading time: 0 Render date: 2026-11-15T22:18:57.763Z Has data issue: false hasContentIssue false
  • English
  • Français

Classes of life distributions and renewal counting process

Part of:Distribution theorySpecial processes

Published online by Cambridge University Press:  14 July 2016

Yi-Hau Chen
Yi-Hau Chen*
Affiliation:
National Taiwan University
*
∗ Postal address: National Taiwan University, College of Public Health, Graduate Institute of Public Health, Division of Biostatistics, No. 1 Jen-Ai Road Sec. 1, Taipei, Taiwan, ROC.
Get accessRights & Permissions [Opens in a new window]

Abstract

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

Keywords

EXPONENTIAL DISTRIBUTIONRENEWAL COUNTING PROCESSDFRNBUNWUNBUENWUE

MSC classification

Primary:62E10: Characterization and structure theory
Secondary:60K10: Applications (reliability, demand theory, etc.)

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 31 , Issue 4 , December 1994, pp. 1110 - 1115
Copyright
Copyright © Applied Probability Trust 1994 

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

Barlow, R. E. and Proschan, F. (1964) Comparison of replacement policies, and renewal theory implications. Ann. Math. Statist. 35, 577–589.Google Scholar
Barlow, R. E. and Proschan, F. (1981) Statistical Theory of Life Testing: Probability Models. 2nd edn. To Begin With, Silver Springs, MD.Google Scholar
Brown, M. (1980) Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes , 2nd edn. Academic Press, New York.Google Scholar
Lau, K. S. and Rao, B. L. C. P. (1990) Characterization of the exponential distribution by the relevation transform. J. Appl. Prob. 27, 726–729.Google Scholar
Marshall, A. W. and Proschan, F. (1972) Classes of distributions applicable in replacement with renewal theory implication. Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 395–415.Google Scholar
Shaked, M. and Zhu, H. (1992) Some results on block replacement policies and renewal theory. J. Appl. Prob. 29, 932–946.Google Scholar