Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T20:55:57.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic comparisons for residual lifetimes and Bayesian notions of multivariate ageing

Part of: Stochastic processes Survival analysis and censored data Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Bruno Bassan  and
Fabio Spizzichino
Bruno Bassan*
Affiliation:
Università ‘La Sapienza’
Fabio Spizzichino*
Affiliation:
Università ‘La Sapienza’
*
∗ Postal address: Dipartimento di Matematica, Università ‘La Sapienza’, Piazzale Aldo Moro 5, I-00185 Roma, Italy.
∗ Postal address: Dipartimento di Matematica, Università ‘La Sapienza’, Piazzale Aldo Moro 5, I-00185 Roma, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

We compare distributions of residual lifetimes of dependent components of different age. This approach yields several notions of multivariate ageing. A special feature of our notions is that they are based on one-dimensional stochastic comparisons. Another difference from the traditional approach is that we do not condition on different histories.

Keywords

Multivariate ageingstochastic orderingsSchur-concavitylog-concavityTP2RR2

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60G09: Exchangeability 90B25: Reliability, availability, maintenance, inspection
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 1078 - 1094
Copyright
Copyright © Applied Probability Trust 1999 

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

Ahmad, R. and Abouammoh, A. M. (1978). Pólya-type, Schur-concave and related probability distributions. In Transactions of the Eighth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes. Reidel, Dordrecht, pp. 57–70 Google Scholar
Arjas, E. (1981a). A stochastic process approach to multivariate reliability systems: notions based on conditional stochastic order. Math. Operat. Res. 6, 263–276.Google Scholar
Arjas, E. (1981b). The failure and hazard processes in multivariate reliability systems. Math. Operat. Res. 6, 551–562.CrossRefGoogle Scholar
Arjas, E. and Norros, I. (1991). Stochastic order and martingale dynamics in multivariate life length models: a review. In Stochastic Orders and Decision under Risk, eds. Mosler, K., Scarsini, M., IMS Lecture Notes Monogr., Hayward, CA, pp. 7–24.Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Barlow, R. E. and Mendel, M. B. (1993). Similarity as a characteristic of wear-out. In Reliability and Decision Making, eds. Barlow, R.E., Clarotti, C.A. and Spizzichino, F., Chapman and Hall, London, pp. 233–245.Google Scholar
Barlow, R. E and Spizzichino, F. (1993). Schur-concave survival functions and survival analysis. J. Comp. Appl. Math. 46, 437–447.Google Scholar
Block, H. W., Savits, T. H. and Shaked, M. (1982). Some concepts of negative dependence. Ann. Prob. 10, 765–772.Google Scholar
Block, H. W., Sampson, A. and Savits, T. H. (eds.) (1990). Topics in Statistical Dependence. IMS Lecture Notes Monogr., Hayward, CA.Google Scholar
Brindley, E. C. and Thompson, W. A. (1972). Dependence and aging aspects of multivariate survival. J. Amer. Statist. Assoc. 67, 822–830.Google Scholar
Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity and Applications. Academic Press, New York.Google Scholar
Ebrahimi, N. and Ghosh, M. (1981). Multivariate negative dependence. Comm. Statist. A 10, 307–337.Google Scholar
Karlin, S. (1968). Total Positivity. Stanford Univeristy Press, Stanford.Google Scholar
Karlin, S. and Rinott, Y. (1980a) Classes of orderings of measures and related correlation inequalities. I Multivariate totally positive distributions. J. Multivariate An. 10, 467–498.Google Scholar
Karlin, S. and Rinott, Y. (1980b). Classes of orderings of measures and related correlation inequalities. II Multivariate reverse rule distributions. J. Multivariate Anal. 10, 499–516.CrossRefGoogle Scholar
Karlin, S. and Rinott, Y. (1988). A generalized Cauchy–Binet formula and applications to total positivity and majorization. J. Multivariate Anal. 27, 284–299.Google Scholar
Marshall, A. V. and Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York.Google Scholar
Prékopa, A., (1973). On logarithmic concave measures and functions. Acta. Sci. Math. (Szeged) 34, 335–343.Google Scholar
Ross, S. M. (1984). A model in which components failure rates depend on the working set. Naval Res. Logist. Quart. 31, 297–301.Google Scholar
Savits, T. H. (1985). A multivariate IFR distribution. J. Appl. Prob. 22, 197–204.Google Scholar
Scarsini, M. and Spizzichino, F. (1999). Simpson-type paradoxes, dependence and ageing. J. Appl. Prob. 36, 119–131.Google Scholar
Shaked, M. (1977). A family of concepts of dependence for bivariate distributions. J. Amer. Statist. Assoc. 72, 642–650.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1988). Multivariate conditional hazard rates and the MIFRA and MIFR properties. J. Appl. Prob. 25, 150–168.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1991). Dynamic multivariate aging notions in reliability theory. Stoch. Proc. Appl. 38, 85–97.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications. Academic Press, London.Google Scholar
Spizzichino, F. (1992). Reliability decision problems under conditions of ageing. In Bayesian Statistics 4, eds. Bernardo, J., Berger, J., Dawid, A.P. and Smith, A.F.M., Clarendon Press, Oxford, pp. 803–811.Google Scholar