Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T12:14:25.910Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the ℒ-class of life distributions

Part of: Operations research and management science Integral transforms, operational calculus Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

B. Klar
B. Klar*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe, Englerstr. 2, 76128 Karlsruhe, Germany. Email address: bernhard.klar@math.uni-karlsruhe.de
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper first recalls some stochastic orderings useful for studying the ℒ-class and the Laplace order in general. We use these orders to show that the higher moments of an ℒ-class distribution need not exist. Using simple sufficient conditions for the Laplace ordering, we give examples of distributions in the ℒ- and ℒα-classes. Moreover, we present explicit sharp bounds on the survival function of a distribution belonging to the ℒ-class of life distributions. The results reveal that the ℒ-class should not be seen as a more comprehensive class of ageing distributions but rather as a large class of life distributions on its own.

Keywords

Laplace orders-(increasing) convex orders-(increasing) concave orderℒα-classreliability boundslife distribution

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Secondary: 44A10: Laplace transform 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 11 - 19
Copyright
Copyright © Applied Probability Trust 2002 

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., and Proschan, F. (1991). Laplace ordering and its applications. J. Appl. Prob. 28, 116–130.Google Scholar
Bhattacharjee, A., and Sengupta, D. (1996). On the coefficient of variation of the ℒ- and ℒ¯-classes. Statist. Prob. Lett. 27, 177–180.Google Scholar
Bhattacharyya, G., and Fries, A. (1982). Fatigue failure models—Birnbaum–Saunders vs. inverse Gaussian. IEEE Trans. Reliab. 31, 439–441.Google Scholar
Chaudhuri, G., Deshpande, J., and Dharmadhikari, A. (1996). A lower bound on the L-class of life distributions and its applications. Calcutta Statist. Assoc. Bull. 46, 269–274.Google Scholar
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
Fishburn, P. (1980a). Continua of stochastic dominance relations for unbounded probability distributions. J. Math. Econom. 7, 271–285.Google Scholar
Fishburn, P. (1980b). Stochastic dominance and moments of distributions. Math. Operat. Res. 5, 94–100.Google Scholar
Henze, N., and Klar, B. (2001). Testing exponentiality against the L-class of life distributions. Math. Methods of Statist. 10, 232–246.Google Scholar
Johnson, N., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Kaas, R., and Hesselager, O. (1995). Ordering claim size distributions and mixed Poisson probabilities. Insurance Math. Econom. 17, 193–201.CrossRefGoogle Scholar
Karlin, S., and Novikoff, A. (1963). Generalized convex inequalities. Pacific J. Math. 13, 1251–1279.Google Scholar
Klefsjö, B. (1982). The HNBUE and HNWUE classes of life distributions. Naval Res. Logistics Quart. 29, 331–344.Google Scholar
Klefsjö, B. (1983). A useful ageing property based on the Laplace transform. J. Appl. Prob. 20, 615–626.CrossRefGoogle Scholar
Kochar, S., and Deshpande, J. (1985). On exponential scores statistic for testing against positive aging. Statist. Prob. Lett. 3, 71–73.Google Scholar
Lin, G. (1998s). Characterizations of the ℒ-class of life distributions. Statist. Prob. Lett. 40, 259–266.Google Scholar
Lin, G. (1998b). On weak convergence within the ℒ-like classes of life distributions. Sankhyā A 60, 176–183.Google Scholar
Lin, G., and Hu, C. (2000). A note on the ℒ-class of life distributions. Sankhyā A 62, 267–272.Google Scholar
Mitra, M., Basu, S., and Bhattacharjee, M. (1995). Characterizing the exponential law under Laplace order domination. Calcutta Statist. Assoc. Bull. 45, 171–178.CrossRefGoogle Scholar
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
Rolski, T. (1975). Mean residual life. Bull. Internat. Statist. Inst. 46, 266–270.Google Scholar
Rolski, T. (1976). Order relations in the set of probability distributions and their applications in queuing theory. Dissertationes Math. 132, 3–47.Google Scholar
Sengupta, D. (1995). Reliability bounds for the ℒ-class and Laplace order. J. Appl. Prob. 32, 832–835.Google Scholar
Stoyan, D. (1983). Comparison Models for Queues and Other Stochastic Models. John Wiley, New York.Google Scholar