Lifetime Data Analysis

, 12:481 | Cite as

An alternative competing risk model to the Weibull distribution for modelling aging in lifetime data analysis

  • Nicolas BousquetEmail author
  • Henri Bertholon
  • Gilles Celeux


A simple competing risk distribution as a possible alternative to the Weibull distribution in lifetime analysis is proposed. This distribution corresponds to the minimum between exponential and Weibull distributions. Our motivation is to take account of both accidental and aging failures in lifetime data analysis. First, the main characteristics of this distribution are presented. Then, the estimation of its parameters are considered through maximum likelihood and Bayesian inference. In particular, the existence of a unique consistent root of the likelihood equations is proved. Decision tests to choose between an exponential, Weibull and this competing risk distribution are presented. And this alternative model is compared to the Weibull model from numerical experiments on both real and simulated data sets, especially in an industrial context.


Failure time distribution Aging Weibull distribution Accidental failure Competing risk model EM algorithm Bayesian inference Importance sampling Likelihood ratio test 



We warmly thank the Associate Editor and the reviewers for their numerous suggestions and corrections which greatly help to improve the presentation of the paper.


  1. d’Agostino RB, Stephens MA (1986) Goodness-of-Fit Techniques. Marcel Dekker, New YorkzbMATHGoogle Scholar
  2. Bacha M, Celeux G, Idée E, Lannoy A, Vasseur D (1998) Estimation de modèles de durées de vie fortement censurées”. Eyrolles, ParisGoogle Scholar
  3. Basu S, Sen A, Banerjee M (2003) Bayesian analysis of competing risks with partially masked cause of failure. Appl Stat 52:77–93MathSciNetzbMATHGoogle Scholar
  4. Berger JO, Sun D (1993) Bayesian analysis for the Poly-Weibull distribution. J Amer Stat Assoc 88:1412–1418CrossRefMathSciNetzbMATHGoogle Scholar
  5. Berger JO, Sun D (1994) Bayesian sequential reliability for Weibull and related distributions. Ann Inst Stat Math 46:221–249CrossRefMathSciNetzbMATHGoogle Scholar
  6. Bertholon H (2001) Une modélisation du vieillissement, PhD Thesis, Université Joseph Fourier, GrenobleGoogle Scholar
  7. Cappé O, Guillin A, Marin JM, Robert CP (2004) Population Monte Carlo. J Comput Graph Stat 13:907–929CrossRefGoogle Scholar
  8. Celeux G, Marin JM, Robert CP (2005) Iterated Importance sampling in Missing Data Problems. Comput Stat Data Anal 50:3386–3404CrossRefMathSciNetGoogle Scholar
  9. Chan V, Meeker WQ (1999) A failure-time model for infant-mortality and wearout failure modes. IEEE Trans Reliab 48:377–387CrossRefGoogle Scholar
  10. Chanda KC (1954) A note on the consistency and maxima of the roots of likelihood equations. Biometrika 41:56–61CrossRefMathSciNetzbMATHGoogle Scholar
  11. Craiu RV, Duchesne T (2004) Inference based on the EM algorithm for the competing risks model with masked causes of failure. Biometrika 91:543–558CrossRefMathSciNetzbMATHGoogle Scholar
  12. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J Roy Stat Soc (Ser B) 39:1–38MathSciNetzbMATHGoogle Scholar
  13. Douc R, Guillin A, Marin JM, Robert CP (2005) Convergence of adaptive sampling schemes. Research Paper RR-5485, INRIA Futurs, FebruaryGoogle Scholar
  14. Erto P (1982) New practical Bayes estimators for the 2-parameter Weibull distribution. IEEE Trans Reliab 31:194–197CrossRefGoogle Scholar
  15. Flehinger BJ, Reiser B, Yashchin E (2002) Parametric modeling for survival with competing risks and masked failure causes. Lifetime Data Anal 8:177–203CrossRefMathSciNetzbMATHGoogle Scholar
  16. Friedman L, Gertsbakh IB (1980) Maximum likelihood estimation in a minimum-type model with exponential and Weibull failure modes. J Amer Stat Assoc 75:460–465CrossRefMathSciNetzbMATHGoogle Scholar
  17. Goetghebeur E, Ryan L (1995) Competing risks survival analysis. Biometrika 42(4):821–833CrossRefMathSciNetGoogle Scholar
  18. Gourieroux Ch, Monfort A (1996) Statistique et modèles économétriques. Economica, ParisGoogle Scholar
  19. INSEE (2001) National French Institute for Statistics and Economic Studies, 195, rue de Bercy—Tour Gamma A—75582 Paris cedex 12, France, Scholar
  20. Lannoy A, Procaccia H (2001) L’utilisation du jugement d’expert en sûreté de fonctionnement, éditions Tec&Doc, ParisGoogle Scholar
  21. Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, New YorkzbMATHGoogle Scholar
  22. McLachlan GJ, Krishnam T (1997). The EM algorithm and extensions. Wiley, New YorkzbMATHGoogle Scholar
  23. Meeker WQ, Escobar LA (1998) Statistical methods for reliability data. New York, WileyzbMATHGoogle Scholar
  24. Park C, Padgett WJ (2004) Analysis of strength distributions of multi-modal failures using the EM algorithm. Technical report No. 220, Department of Statistics, University of South Carolina,Google Scholar
  25. Robert CP, Casella G (1999) Monte Carlo statistical methods. Springer-Verlag, New YorkzbMATHGoogle Scholar
  26. Schafer RE (1969) Bayesian reliability demonstration, phase I–data for the a priori distribution, RADC-TR-69-389, Rome Air Development CenterGoogle Scholar
  27. Schafer RE, Sheffield TS (1971) Bayesian reliability demonstration, phase II—development of a prior distribution, RADC-TR-71-139, Rome Air Development CenterGoogle Scholar
  28. Steele R, Raftery AE, Emond M (2003) Computing normalizing constants for finite mixture models via incremental mixture importance sampling (IMIS). Technical Report no. 436, Department of Statistics, University of WashingtonGoogle Scholar
  29. Tanner MA, Wong W (1987) The calculation of posterior distributions by data augmentation (with discussion). J Amer Stat Assoc 82:528–550CrossRefMathSciNetzbMATHGoogle Scholar
  30. Tanner MA (1991) Tools for statistical inference, observed data and data augmentation methods, Lectures notes in statistics. Springer-Verlag, New YorkGoogle Scholar
  31. Usher JS, Hodgson TJ (1988) Maximum Likelihood analysis of component reliability using masked system life data. IEEE Trans on Reliab 37:550–555CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Nicolas Bousquet
    • 1
    Email author
  • Henri Bertholon
    • 2
    • 3
  • Gilles Celeux
    • 1
  1. 1.INRIA FutursUniversité Paris-SudOrsayFrance
  2. 2.CNAM ParisParisFrance
  3. 3.INRIA Rhone-AlpesMontbonnot Saint MartinFrance

Personalised recommendations