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Some bivariate notions of IFR and DMRL and related properties

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

Published online by Cambridge University Press:  14 July 2016

Bruno Bassan ,
Subhash Kochar  and
Fabio Spizzichino
Bruno Bassan*
Affiliation:
Università di Roma ‘La Sapienza’
Subhash Kochar*
Affiliation:
Indian Statistical Institute
Fabio Spizzichino*
Affiliation:
Università di Roma ‘La Sapienza’
*
∗ Postal address: Dipartimento di Matematica, Università di Roma ‘La Sapienza’, Piazzale Aldo Moro 5, 00185 Roma, Italy.
∗∗∗ Postal address: Indian Statistical Institute, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India.
∗ Postal address: Dipartimento di Matematica, Università di Roma ‘La Sapienza’, Piazzale Aldo Moro 5, 00185 Roma, Italy.
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Abstract

Recently Bassan and Spizzichino (1999) have given some new concepts of multivariate ageing for exchangeable random variables, such as a special type of bivariate IFR, by comparing distributions of residual lifetimes of dependent components of different ages. In the same vein, we further study some properties of these concepts of IFR in the bivariate case. Then we introduce certain concepts of bivariate DMRL ageing and we develop a treatment that parallels those developed for bivariate IFR. For both the IFR and DMRL concepts, we analyse a weak and a strong version, and discuss some of the differences between them.

Keywords

Residual lifetimemean residual life functionmultivariate extensions of IFR and DMRLlog-concavitySchur-concavityageing properties of mixtures

MSC classification

Primary: 62N05: Reliability and life testing 60G09: Exchangeability 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 533 - 544
Copyright
Copyright © Applied Probability Trust 2002 

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References

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
Barlow, R. E., and Mendel, M. B. (1992). De Finetti-type representations for life distributions. J. Amer. Statist. Assoc. 87, 1116–1122.Google Scholar
Barlow, R. E., and Mendel, M. B. (1993). Similarity as a characteristic of ageing. In Reliability and Decision Making, eds Barlow, R. E., Clarotti, C. A. and Spizzichino, F., Chapman and Hall, London, pp. 233–245.CrossRefGoogle Scholar
Barlow, R., and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD.Google Scholar
Barlow, R. E., and Spizzichino, F. (1993). Schur-concave survival functions and survival analysis. J. Comput. Appl. Math. 46, 437–447.Google Scholar
Bassan, B., and Spizzichino, F. (1999). Stochastic comparison for residual lifetimes and Bayesian notions of multivariate ageing. Adv. Appl. Prob. 31, 1078–1094.CrossRefGoogle Scholar
Brindley, E. C., and Thompson, W. A. (1972). Dependence and aging aspects of multivariate survival. J. Amer. Statist. Assoc. 67, 822–830.CrossRefGoogle Scholar
Deshpande, J. V., Kochar, S. C., and Singh, H. (1986). Aspects of positive ageing. J. Appl. Prob. 23, 748–58.CrossRefGoogle Scholar
Finkelstein, M. S., and Esaulova, V. (2001). Modeling a failure rate for a mixture of distribution functions. Prob. Eng. Inf. Sci. 15, 383–400.Google Scholar
Guess, F., and Proschan, F. (1988). Mean residual life: theory and applications. In Handbook of Statistics, Vol. 7, Quality Control and Reliability, eds Krishnaiah, P. R. and Rao, C. R., North-Holland, Amsterdam, pp. 215–224.Google Scholar
Karlin, S. (1968). Total Positivity. Stanford University Press.Google Scholar
Khaledi, B., and Kochar, S. (2000). Stochastic comparisons and dependence among concomitants of order statistics. J. Multivariate Anal. 73, 262–281.Google Scholar
Kochar, S., Mukerjee, H., and Samaniego, F. (2000). Estimation of a monotone mean residual life. Ann. Statist. 28, 905–921.Google Scholar
Lynch, J. D. (1999). On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate. Prob. Eng. Inf. Sci. 13, 33–36.CrossRefGoogle Scholar
Marshall, A. W., 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
Savits, T. H. (1985). A multivariate IFR distribution. J. Appl. Prob. 22, 197–204.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. Process. Appl. 38, 85–97.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, San Diego, CA.Google Scholar
Shaked, M., and Spizzichino, F. (2001). Mixtures and monotonicity of failure rate functions. In Handbook of Statistics, Vol. 20, Advances in Reliability, eds Balakrishnan, N. and Rao, C. R., North-Holland, Amsterdam, pp. 185–198.Google Scholar
Spizzichino, F. (2001). Subjective Probability Models for Lifetimes. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar