Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T04:48:21.401Z Has data issue: false hasContentIssue false
  • English
  • Français

Multivariate conditional hazard rates and the MIFRA and MIFR properties

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, The University of Arizona, Tucson, AZ 85721, USA.
∗∗ Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

If T = (T1, · ··, Tn) is a vector of random lifetimes then its distribution can be determined by a set λof multivariate conditional hazard rates. In this paper, sufficient conditions on λare found which imply that T is Block–Savits MIFRA (multivariate increasing failure rate average) or Savits MIFR (multivariate increasing failure rate). Applications for a multivariate reliability model of Ross and for load-sharing models are given. The relationship between Shaked and Shanthikumar model of multivariate imperfect repair and the MIFRA property is also discussed.

Keywords

MULTIVARIATE IFR AND IFRALOGCONCAVITYSUBHOMOGENEOUS AND SUPERHOMO-GENEOUS FUNCTIONSRELIABILITY MODELINGLOAD-SHARING MODELINGMULTIVARIATE IMPERFECT REPAIR
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 150 - 168
Copyright
Copyright © Applied Probability Trust 1988 

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 Air Force Office of Scientific Research, USAF, under Grant AFOSR-84–0205. Reproduction in whole or in part is permitted for any purpose of the United States Government.

References

Aalen, O. O. and Hoem, J. M. (1978) Random times changes for multivariate counting processes. Scand. Actuarial J. , 81–101.Google Scholar
Arjas, E. (1981a) A stochastic process approach to multivariate reliability: notions based on conditional stochastic order. Math. Operat. Res. 6, 263–276.CrossRefGoogle Scholar
Arjas, E. (1981b) The failure and hazard processes in multivariate reliability systems. Math. Operat. Res. 6, 551–562.CrossRefGoogle Scholar
Berg, M. and Cleroux, R. (1982) A marginal cost analysis for an age replacement policy with minimal repair. INFOR 20, 258–263.Google Scholar
Bhattacharjee, M. C. (1984) New results for the Brown–Proschan model of imperfect repair. Technical Report, Indian Institute of Management, Calcutta, India.Google Scholar
Block, H. W. and Savits, T. H. (1980) Multivariate IFRA distributions. Ann. Prob. 8, 793–801.Google Scholar
Brémaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer-Verlag, New York.Google Scholar
Brown, M. and Proschan, F. (1983) Imperfect repair. J. Appl. Prob. 20, 851–859.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, 1966–1974.Google Scholar
Jacobsen, M. (1982) Statistical Analysis of Counting Processes , Lecture Notes in Statistics 12, Springer-Verlag, New York.Google Scholar
Kurtz, T. G. (1980) Representations of Markov processes as multiparameter time changes. Ann Prob. 8, 682–715.Google Scholar
Lawless, J. F. (1982) Statistical Models and Methods for Lifetime Data. Wiley, New York.Google Scholar
Marshall, A. W. and Shaked, M. (1982) A class of multivariate new better than used distributions. Ann. Prob. 10, 259–264.CrossRefGoogle Scholar
Norros, I. (1986) A compensator representation of multivariate lifelength distributions, with applications. Scand. J. Statist , 13, 99–112.Google Scholar
Ross, S. M. (1984) A model in which component failure rates depend on the working set. Naval Res. Logist. Quart. 31, 297–300.Google Scholar
Savits, T. H. (1985) A multivariate IFR class. J. Appl. Prob. 22, 197–204.Google Scholar
Schechner, Z. (1984) A load-sharing model: the linear breakdown rule. Naval Res. Log. Quart. 31, 137–144.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1986a) Multivariate imperfect repair. Operat. Res. 34, 437–448.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1986b) The total hazard construction, antithetic variates and simulation of stochastic systems. Stochastic Models 2, 237–249.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1987a) Multivariate hazard rates and stochastic ordering. Adv. Appl. Prob. 19, 123–137.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1987b) The multivariate hazard construction. Stoch. Proc. Appl. 24, 241–258.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Stochastic Models. Wiley, New York.Google Scholar