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Characterizations of classes of lifetime distributions generalizing the NBUE class

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

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: bernhard.klar@math.uni-karlsruhe.de
∗∗ Postal address: Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Germany.
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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 

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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