Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T11:26:00.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic multivariate mean residual life functions

Published online by Cambridge University Press:  14 July 2016

Moshe Shaked  and
J. George Shanthikumar
Moshe Shaked*
Affiliation:
University of Arizona
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Mathematics, Building #89, Universify of Arizona, Tucson, AZ 85721, USA.
∗∗ Postal address: Management Science Group, W. A. Haas School of Business, University of California, Berkeley, CA 94720, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied.

A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given.

Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.

Keywords

DECREASING MEAN RESIDUAL LIFEHAZARD RATE ORDERINGNOTIONS OF POSITIVE DEPENDENCESTOCHASTIC ORDERINGMULTIVARIATE AGING NOTIONSRELIABILITY THEORYWEAKENED BY FAILUREASSOCIATION OF RANDOM VARIABLESFREUND DISTRIBUTIONSTOCHASTIC MODELING
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 613 - 629
Copyright
Copyright © Applied Probability Trust 1991 

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.)

Footnotes

Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant ADOSR-84–0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.

References

Ahmed, A.-H. N. (1988) Preservation properties for the mean residual life ordering. Statist. Papers 29, 143–150.Google Scholar
Azaid, A. A. (1988) Mean residual life ordering. Statist. Papers 29, 35–43.Google Scholar
Arjas, E. (1981) A stochastic process approach to multivariate reliability systems: Notions based on conditional stochastic order. Math. Operat. Res. 6, 263–276.Google Scholar
Arjas, E. and Norros, I. (1984) Life lengths and associations: A dynamic approach. Math. Operat. Res. 9, 151–158.CrossRefGoogle Scholar
Arnold, B. C. and Zahedi, H. (1988) On multivariate mean remaining life functions. J. Multivariate Anal. 25, 1–9.Google Scholar
Baccelli, F. and Makowski, A. M. (1989) Multidimensional stochastic ordering and associated random variables. Operat. Res. 37, 478–487.Google Scholar
Bhattacharjee, M. C. (1982) The class of mean residual lives and some consequences. SIAM J. Alg. Disc. Meth. 3, 56–65.CrossRefGoogle Scholar
Buchanan, W. B. and Singpurwalla, N. D. (1977) Some stochastic characterizations of multivariate survival, In Theory and Applications of Reliability, Vol. 1, ed. Tsokos, C. P. and Shimi, I. N., Academic Press, New York, pp. 329–348.Google Scholar
Cox, D. R. (1972) Regression models and life tables (with discussion). J. R. Statist. Soc. B 34, 187–202.Google Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.Google Scholar
Freund, J. E. (1961) A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971–977.Google Scholar
Guess, F. and Proschan, F. (1988) Mean residual life: theory and applications. In Handbook of Statistics, Vol. 7, ed. Krishnaiah, P. R. and Rao, C. R., Elsevier, Amsterdam, 215–224.Google Scholar
Gupta, R. C. (1975) On characterization of distributions by conditional expectations. Commun. Statist. 4, 99–103.CrossRefGoogle Scholar
Hall, W. J. and Wellner, J. A. (1981) Mean residual life. In Statistics and Related Topics, eds. Csorgo, M., Dawson, D. A., Rao, J. N. K. and Saleh, A. K. M. E., North-Holland, Amsterdam, pp. 169–184.Google Scholar
Kochar, S. C. (1989) On extensions of DMRL and related partial orderings of life distributions. Commun. Statist.–Stoch. Models 5, 235–245.Google Scholar
Kochar, S. C. and Wiens, D. P. (1987) Partial orderings of life distributions with respect to their aging properties. Naval Res. Logist. 34, 823–829.Google Scholar
Nair, K. R. and Nair, N. V. (1989) Bivariate mean residual life. IEEE Trans. Reliability 38, 362–364.Google Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Ross, S. M. (1984) A model in which component failure rates depend on the working set. Naval Res. Logist. Quart. 31, 297–301.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1986) Multivariate imperfect repair. Operat. Res. 34, 437–448.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1987) Multivariate hazard rates and stochastic ordering. Adv. Appl. Prob. 19, 123–137.Google Scholar
Shared, M. and Shanthikumar, J. G. (1991) Multivariate increasing failure rate and PF2 distributions in reliability theory. Stoch. Proc. Appl. To appear.Google Scholar
Shared, M. and Shanthikumar, J. G. (1990) Multivariate stochastic orderings and positive dependence in reliability theory. Math. Oper. Res. 15, 545–552.Google Scholar
Zahedi, H. (1985) Some new classes of multivariate survival distribution functions. J. Statist. Planning Inf. 11, 171–188.Google Scholar