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Asymptotic behavior of a general class of mixture failure rates

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Maxim Finkelstein  and
Veronica Esaulova
Maxim Finkelstein*
Affiliation:
University of the Free State, Bloemfontein, and Max Planck Institute for Demographic Research
Veronica Esaulova*
Affiliation:
Otto-von-Guericke-Universität Magdeburg
*
∗ Postal address: Department of Mathematical Statistics, University of the Free State, PO Box 339, 9300 Bloemfontein, Republic of South Africa. Email address: [email protected]
∗∗ Postal address: Faculty of Mathematics, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany. Email address: [email protected]
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Abstract

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Mixtures of increasing failure rate distributions can decrease, at least in some intervals of time. Usually this property is observed asymptotically, as t → ∞, which is due to the fact that a mixture failure rate is ‘bent down’, as the weakest populations are dying out first. We consider a survival model that generalizes additive hazards models, proportional hazards models, and accelerated life models very well known in reliability and survival analysis. We obtain new explicit asymptotic relations for a general setting and study specific cases. Under reasonable assumptions we prove that the asymptotic behavior of the mixture failure rate depends only on the behavior of the mixing distribution in the neighborhood of the left-hand endpoint of its support, and not on the whole mixing distribution.

Keywords

Mixture of distributionsdecreasing failure rateincreasing failure rateproportional hazards modelaccelerated life model

MSC classification

Primary: 62E10: Characterization and structure theory 60E15: Inequalities; stochastic orderings
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 244 - 262
Copyright
Copyright © Applied Probability Trust 2006 

References

Anderson, J. E. and Louis, T. A. (1995). Survival analysis using a scale change random effects model. J. Amer. Statist. Assoc. 90, 669–679.Google Scholar
Barlow, R. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Block, H. W. and Joe, H. (1997). Tail behavior of the failure rate functions of mixtures. Lifetime Data Anal. 3, 269–288.Google Scholar
Block, H. W., Li, Y. and Savits, T. H. (2003). Initial and final behavior of failure rate functions for mixtures and systems. J. Appl. Prob. 40, 721–740.Google Scholar
Block, H. W., Mi, J. and Savits, T. H. (1993). Burn-in and mixed populations. J. Appl. Prob. 30, 692–702.Google Scholar
Carey, J. R., Liedo, P., Orozco, D. and Vaupel, J. W. (1992). Slowing of mortality rates at older ages in large medfly cohorts. Science 258, 457–461.CrossRefGoogle ScholarPubMed
Clarotti, C. A. and Spizzichino, F. (1990). Bayes burn-in decision procedures. Prob. Eng. Inf. Sci. 4, 437–445.Google Scholar
Finkelstein, M. S. (2004). Minimal repair in heterogeneous populations. J. Appl. Prob. 41, 281–286.Google Scholar
Finkelstein, M. S. (2005). On some reliability approaches to human aging. Internat. J. Reliability Quality Safety Eng. 12, 1–10.CrossRefGoogle Scholar
Finkelstein, M. S. and Esaulova, V. (2001a). Modeling a failure rate for a mixture of distribution functions. Prob. Eng. Inf. Sci. 15, 383–400.Google Scholar
Finkelstein, M. S. and Esaulova, V. (2001b). On an inverse problem in mixture hazard rates modelling. Appl. Stoch. Models Business Industry 17, 221–229.Google Scholar
Gupta, P. L. and Gupta, R. C. (1996). Ageing characteristics of the Weibull mixtures. Prob. Eng. Inf. Sci. 10, 591–600.Google Scholar
Gurland, J. and Sethuraman, J. (1995). How pooling failure data may reverse increasing failure rate. J. Amer. Statist. Assoc. 90, 1416–1423.Google Scholar
Li, Y. (2005). Asymptotic baseline of the hazard rate function of mixtures. J. Appl. Prob. 42, 892–901.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.Google Scholar
Lynn, N. J. and Singpurwalla, N. D. (1997). Comment: “Burn-in” makes us feel good. Statist. Sci. 12, 13–19.Google Scholar
Thatcher, A. R. (1999). The long-term pattern of adult mortality and the highest attained age. J. R. Statist. Soc. A 162, 5–43 (with discussion).Google Scholar
Vaupel, J. W., Manton, K. G. and Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16, 439–454.Google Scholar