Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T23:10:52.171Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of classes of lifetime distributions generalizing the NBUE class

Part of: Operations research and management science Survival analysis and censored data Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Bernhard Klar  and
Alfred Müller
Bernhard Klar*
Affiliation:
Universität Karlsruhe
Alfred Müller*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany. Email address: [email protected]
∗∗ Postal address: Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a new class of lifetime distributions exhibiting a notion of positive ageing, called the ℳ-class, which is strongly related to the well-known ℒ-class. It is shown that distributions in the ℳ-class cannot have an undesirable property recently observed in an example of an ℒ-class distribution by Klar (2002). Moreover, it is shown how these and related classes of life distributions can be characterized by expected remaining lifetimes after a family of random times, thus extending the notion of NBUE. We give examples of ℳ-class distributions by using simple sufficient conditions, and we derive reliability bounds for distributions in this class.

Keywords

Ageingℒ-classℳ-classNBUEreliability boundsstochastic orderingmoment generating function

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60E15: Inequalities; stochastic orderings 62E10: Characterization and structure theory 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 20 - 32
Copyright
Copyright © Applied Probability Trust 2003 

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

[1] Alzaid, A., Kim, J. S., and Proschan, F. (1991). Laplace ordering and its applications. J. Appl. Prob. 28, 116–130.Google Scholar
[2] Barlow, R. E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, New York.Google Scholar
[3] Chhikara, R., and Folks, J. (1989). The Inverse Gaussian Distribution, Theory, Methodology and Applications. Marcel Dekker, New York.Google Scholar
[4] Denuit, M. (2001). Laplace transform ordering of actuarial quantities. Insurance Math. Econom. 29, 83–102.CrossRefGoogle Scholar
[5] Denuit, M., Lefevre, C., and Shaked, M. (1998). The s-convex orders among real random variables, with applications. Math. Inequal. Appl. 1, 585–613.Google Scholar
[6] Deshpande, J. V., Kochar, S. C., and Singh, H. (1986). Aspects of positive ageing. J. Appl. Prob. 23, 748–758.Google Scholar
[7] Fagiuoli, E., and Pellerey, F. (1993). New partial orderings and applications. Naval Res. Logistics 40, 829–842.3.0.CO;2-D>CrossRefGoogle Scholar
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
[9] Fishburn, P. C. (1980). Continua of stochastic dominance relations for unbounded probability distributions. J. Math. Econom. 7, 271–285.CrossRefGoogle Scholar
[10] Johnson, N., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
[11] Kaas, R., and Hesselager, O. (1995). Ordering claim size distributions and mixed Poisson probabilities. Insurance Math. Econom. 17, 193–201.Google Scholar
[12] Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.CrossRefGoogle Scholar
[13] Klar, B. (2002). A note on the ℒ-class of life distributions. J. Appl. Prob. 39, 11–19.CrossRefGoogle Scholar
[14] Klefsjö, B. (1982). The HNBUE and HNWUE classes of life distributions. Naval Res. Logistics Quart. 29, 331–344.CrossRefGoogle Scholar
[15] Klefsjö, B. (1983). A useful ageing property based on the Laplace transform. J. Appl. Prob. 20, 615–626.Google Scholar
[16] Kotlyar, V., and Khomenko, L. (1993). Generalization of NBUAFR properties of distribution functions. Cybernetics Systems Anal. 29, 735–746.CrossRefGoogle Scholar
[17] Lin, G. D. (1998). Characterizations of the ℒ-class of life distributions. Statist. Prob. Lett. 40, 259–266.Google Scholar
[18] Lukacs, E. (1970). Characteristic Functions, 2nd edn. Hafner, New York.Google Scholar
[19] Müller, A. (1997). Stochastic orders generated by integrals: a unified study. Adv. Appl. Prob. 29, 414–428.Google Scholar
[20] Müller, A., and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
[21] Reuter, H., and Riedrich, T. (1981). On maximal sets of functions compatible with a partial ordering for distribution functions. Math. Operationsforsch. Statist. Ser. Optimization 12, 597–605.CrossRefGoogle Scholar
[22] Rolski, T. (1975). Mean residual life. Bull. Inst. Internat. Statist. 46, 266–270.Google Scholar
[23] Rolski, T. (1976). Order relations in the set of probability distribution functions and their applications in queueing theory. Dissertationes Math. (Rozprawy Mat.) 132, 1–52.Google Scholar
[24] Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston.Google Scholar
[25] Stein, W., and Dattero, R. (1999). Bondesson's functions in reliability theory. Appl. Stoch. Models Business Industry 15, 103–109.3.0.CO;2-C>CrossRefGoogle Scholar
[26] Widder, D. V. (1941). The Laplace Transform. Princeton University Press.Google Scholar