Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-27T18:28:29.203Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic failure rates for a general class of frailty models

Part of: Distribution theory Survival analysis and censored data

Published online by Cambridge University Press:  17 November 2017

Ramesh C. Gupta  and
David M. Bradley
Ramesh C. Gupta*
Affiliation:
University of Maine
David M. Bradley*
Affiliation:
University of Maine
*
* Postal address: Department of Mathematics & Statistics, University of Maine, 5752 Neville Hall, Orono, ME 04469-5752, USA.
* Postal address: Department of Mathematics & Statistics, University of Maine, 5752 Neville Hall, Orono, ME 04469-5752, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We elucidate the long-term behavior of failure rates for a broad class of frailty models in survival analysis. The class properly includes the proportional hazard frailty model, the additive frailty model, and the accelerated failure time frailty model. A complete asymptotic expansion is derived and compared with the corresponding result for the limiting behavior obtained by Finkelstein and Esaulova (2006a). Several examples are provided to facilitate the comparison and to illustrate both the applicability and the limitations of our approach.

Keywords

Mixtureproportional hazard modeladditive hazard modelaccelerated failure time modelfrailty modelmonotonicitymixing distributionsasymptotic expansion

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 62E20: Asymptotic distribution theory
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017 , pp. 1230 - 1259
Copyright
Copyright © Applied Probability Trust 2017 

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

Badía, F. G., Berrade, M. D. and Campos, C. A. (2002). Aging properties of the additive and proportional hazard mixing models. Reliab. Eng. System Safety 78, 165–172. Google Scholar
Block, H. and Joe, H. (1997). Tail behavior of the failure rate functions of mixtures. Lifetime Data Analysis 3, 269–288. Google Scholar
Block, H. W., Li, Y. and Savits, T. H. (2003). Initial and final behaviour of failure rate functions for mixtures and systems. J. Appl. Prob. 40, 721–740. CrossRefGoogle Scholar
Block, H. W., Li, Y., Savits, T. H. and Wang, J. (2008). Continuous mixtures with bathtub-shaped failure rates. J. Appl. Prob. 45, 260–270. Google Scholar
Bogachev, V. I. (2007). Measure Theory. Springer, Berlin. Google Scholar
Bradley, D. M. and Gupta, R. C. (2003). Limiting behaviour of the mean residual life. Ann. Inst. Statist. Math. 55, 217–226. Google Scholar
Chaubey, Y. P. and Sen, P. K. (1999). On smooth estimation of mean residual life. J. Statist. Planning Inference 75, 223–236. CrossRefGoogle Scholar
Cocozza-Thivent, C. and Kalashnikov, V. (1996). The failure rate in reliability: approximations and bounds. J. Appl. Math. Stoch. Analysis 9, 497–530. Google Scholar
Cocozza-Thivent, C. and Kalashnikov, V. (1997). The failure rate in reliability: numerical treatment. J. Appl. Math. Stoch. Analysis 10, 21–45. CrossRefGoogle Scholar
Cohn, D. L. (2013). Measure Theory, 2nd edn. Birkhäuser, New York. 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. CrossRefGoogle Scholar
Finkelstein, M. S. and Esaulova, V. (2001b). On an inverse problem in mixture failure rates modelling. Appl. Stoch. Models Business Industry 17, 221–229. CrossRefGoogle Scholar
Finkelstein, M. and Esaulova, V. (2006a). Asymptotic behavior of a general class of mixture failure rates. Adv. Appl. Prob. 38, 244–262. Google Scholar
Finkelstein, M. and Esaulova, V. (2006b). On mixture failure rate ordering. Commun. Statist. Theory Meth. 35, 1943–1955. CrossRefGoogle Scholar
Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series. Cambridge University Press. Google Scholar
Gordon, R. A. (1994). The Integrals of Lebesgue, Denjoy, Perron, and Henstock (Grad. Studies Math. 4). American Mathematical Society, Providence, RI. CrossRefGoogle Scholar
Grall, A., Dieulle, L., Bérenguer, C. and Roussignol, M. (2006). Asymptotic failure rate of a continuously monitored system. Reliab. Eng. System Safety 91, 126–130. Google Scholar
Gupta, R. C. (2016). Properties of additive frailty model in survival analysis. Metrika 79, 1–17. Google Scholar
Gupta, R. C. and Bradley, D. M. (2003). Representing the mean residual life in terms of the failure rate. Math. Comput. Modelling 37, 1271–1280. Google Scholar
Gupta, R. C. and Gupta, R. D. (2009). General frailty model and stochastic orderings. J. Statist. Planning Inference 139, 3277–3287. Google Scholar
Gupta, R. C. and Gupta, R. D. (2010). Random effect bivariate survival models and stochastic comparisons. J. Appl. Prob. 47, 426–440. Google Scholar
Gupta, R. C. and Kirmani, S. N. U. A. (2006). Stochastic comparisons in frailty models. J. Statist. Planning Inference 136, 3647–3658. Google Scholar
Gupta, R. C. and Warren, R. (2001). Determination of change points of non-monotonic failure rates. Commun. Statist. Theory Meth. 30, 1903–1920. CrossRefGoogle Scholar
Gurland, J. and Sethuraman, J. (1994). Reversal of increasing failure rates when pooling failure data. Technometrics 36, 416–418. Google Scholar
Gurland, J. and Sethuraman, J. (1995). How pooling failure data may reverse increasing failure rates. J. Amer. Statist. Assoc. 90, 1416–1423. Google Scholar
Keiding, N., Andersen, P. K. and Klein, J. P. (1997). The role of frailty models and accelerated failure time models in describing heterogeneity due to omitted covariates. Statist. Medicine 16, 215–224. 3.0.CO;2-J>CrossRefGoogle ScholarPubMed
Lambert, P., Collett, D., Kimber, A. and Johnson, R. (2004). Parametric accelerated failure time models with random effects and an application to kidney transplant survival. Statist. Medicine 23, 3177–3192. Google Scholar
Li, Y. (2005). Asymptotic baseline of the hazard rate function of mixtures. J. Appl. Prob. 42, 892–901. CrossRefGoogle Scholar
Liang, K.-Y., Self, S. G., Bandeen-Roche, K. J. and Zeger, S. L. (1995). Some recent developments for regression analysis of multivariate failure time data. Lifetime Data Analysis 1, 403–415. Google Scholar
Lin, D. Y., Oakes, D. and Ying, Z. (1998). Additive hazards regression with current status data. Biometrika 85, 289–298. Google Scholar
Lin, D. Y. and Ying, Z. (1994). Semiparametric analysis of the additive risk model. Biometrika 81, 61–71. Google Scholar
Magnus, W. and Oberhettinger, F. (1949). Formulas and Theorems for the Special Functions of Mathematical Physics. Chelsea, New York. Google Scholar
Martinussen, T. and Scheike, T. H. (2002). Efficient estimation in additive hazards regression with current status data. Biometrika 89, 649–658. Google Scholar
Martinussen, T., Scheike, T. H. and Zucker, D. M. (2011). The Aalen additive gamma frailty hazards model. Biometrika 98, 831–843. Google Scholar
Mercier, S. and Roussignol, M. (2003). Asymptotic failure rate of a Markov deteriorating system with preventive maintenance. J. Appl. Prob. 40, 1–19. Google Scholar
Missov, T. I. and Finkelstein, M. (2011). Admissible mixing distributions for a general class of mixture survival models with known asymptotics. Theoret. Pop. Biol. 80, 64–70. CrossRefGoogle ScholarPubMed
Nair, N. U. and Sankaran, P. G. (2012). Some results on an additive hazards model. Metrika 75, 389–402. Google Scholar
Pan, W. (2001). Using frailties in the accelerated failure time model. Lifetime Data Analysis 7, 55–64. Google Scholar
Ramanujan, S. (1915). Some definite integrals. Mess. Math. 44, 10–18. 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