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On the collapsibility of lifetime regression models

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

Published online by Cambridge University Press:  01 July 2016

Thierry Duchesne  and
Jeffrey S. Rosenthal
Thierry Duchesne*
Affiliation:
Université Laval, Québec
Jeffrey S. Rosenthal*
Affiliation:
University of Toronto
*
∗ Postal address: Département de Mathématiques et de Statistique, Pavillon Alexandre-Vachon, Université Laval, Québec, Québec G1K 7P4, Canada. Email address: duchesne@mat.ulaval.ca
∗∗ Postal address: Department of Statistics, University of Toronto, 100 St. George Street, Toronto, Ontario M5S 3G3, Canada.
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Abstract

In this paper we derive conditions on the internal wear process under which the resulting time to failure model will be of the simple collapsible form when the usage accumulation history is available. We suppose that failure occurs when internal wear crosses a certain threshold or a traumatic event causes the item to fail. We model the infinitesimal increment in internal wear as a function of time, accumulated internal wear, and usage history, and we derive conditions on this function to get a collapsible model for the distribution of time to failure given the usage history. We reach the conclusion that collapsible models form the subset of accelerated failure time models with time-varying covariates for which the time transformation function satisfies certain simple properties.

Keywords

Accelerated failure time modeladditive hazard modelcollapsible modeldegradationdifferential equationdiffusiongamma processinternal weartraumatic eventusage rate

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 62N05: Reliability and life testing 90B25: Reliability, availability, maintenance, inspection
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 755 - 772
Copyright
Copyright © Applied Probability Trust 2003 

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References

Aalen, O. O. and Gjessing, H. K. (2001). Understanding the shape of the hazard rate: a process point of view. Statist. Sci. 16, 1–22.Google Scholar
Bagdonavicius, V. B. and Nikulin, M. S. (1997). Transfer functionals and semiparametric regression models. Biometrika 84, 365–378.Google Scholar
Bagdonavicius, V. B. and Nikulin, M. S. (2001). Estimation in degradation models with explanatory variables. Lifetime Data Anal. 7, 85–103.CrossRefGoogle ScholarPubMed
Bhattacharya, R. N. and Waymire, E. C. (1990). Stochastic Processes With Applications. John Wiley, New York.Google Scholar
Braun, M. (1983). Differential Equations and Their Applications, 3rd edn. Springer, New York.Google Scholar
Çinlar, E. and Ozekici, S. (1987). Reliability of complex devices in random environments. Prob. Eng. Inf. Sci. 1, 97–115.CrossRefGoogle Scholar
Cox, D. R. (1999). Some remarks on failure-times, surrogate markers, degradation, wear and the quality of life. Lifetime Data Anal. 5, 307–314.CrossRefGoogle ScholarPubMed
Cox, D. R. and Oakes, D. (1984). Analysis of Survival Data. Chapman and Hall, London.Google Scholar
Duchesne, T. (2000). Semiparametric methods of time scale selection. In Recent Advances in Reliability Theory: Methodology, Practice and Inference, eds Limnios, N. and Nikulin, M., Birkhäuser, Boston, MA, pp. 279–290.Google Scholar
Duchesne, T. and Lawless, J. (2000). Alternative time scales and failure time models. Lifetime Data Anal. 6, 157–179.Google Scholar
Durrett, R. (1991). Probability: Theory and Examples. Wadsworth and Brooks/Cole, Belmont, CA.Google Scholar
Finkelstein, M. S. (1999). Wearing-out of components in a variable environment. Reliab. Eng. System Safety 66, 235–242.CrossRefGoogle Scholar
Jewell, N. P. and Kalbfleisch, J. D. (1996). Marker processes in survival analysis. Lifetime Data Anal. 2, 15–29.Google Scholar
Kordonsky, K. B. and Gertsbakh, I. (1997). Multiple time scales and the lifetime coefficient of variation: engineering applications. Lifetime Data Anal. 3, 139–156.CrossRefGoogle ScholarPubMed
Marsden, J. E. (1974). Elementary Classical Analysis. Freeman, New York.Google Scholar
Nelson, W. (1990). Accelerated Testing: Statistical Models, Test Plans and Data Analyses. John Wiley, New York.CrossRefGoogle Scholar
Oakes, D. (1995). Multiple time scales in survival analysis. Lifetime Data Anal. 1, 7–18.CrossRefGoogle ScholarPubMed
Robins, J. and Tsiatis, A.A. (1992). Semiparametric estimation of an accelerated failure time model with time-dependent covariates. Biometrika 79, 311–319.Google Scholar
Rosenthal, J. S. (2000). A First Look at Rigorous Probability Theory. World Scientific, Singapore.Google Scholar
Singpurwalla, N. D. (1995). Survival in dynamic environments. Statist. Sci. 10, 86–103.CrossRefGoogle Scholar
Singpurwalla, N. D. and Wilson, S. P. (1998). Failure models indexed by two scales. Adv. Appl. Prob. 30, 1058–1072.Google Scholar
Wenocur, M. L. (1989). A reliability model based on the gamma process and its analytic theory. Adv. Appl. Prob. 21, 899–918.CrossRefGoogle Scholar