Empirical Hazard Function Using Continuous-Time Failure Data

  • Gang XieEmail author
  • Fengfeng Li
  • Yong Sun
  • Lin Ma
Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)


A theoretically sound and accurate empirical hazard function may be used directly for analysis of lifetime distribution of the continuous-time failure data or can be used as a basis for further parametric modeling analysis in asset management. For the sake of bridging the gaps between probability theory and data analysis practice, this paper starts from clarifying the relationship between the concepts of hazard function and failure rate. Then, two often-used continuous-time data empirical hazard function formulas are derived directly from discrediting their theoretic definitions of the hazard function. The properties of these two different formulas are investigated, and their estimation performances against the true hazard function values are compared using simulation samples from exponential and Weibull distributions. Under the specified assumption conditions, both theoretic calculation and simulation results show that the formula that calculates the average failure rates (AFR) gives less biased estimation than the other one in all cases we have examined. We also showed that under practical situations, the relative difference of the calculated empirical hazards between these two formulas is less than 6 %. Based on the result of this study, we proposed a rule of thumb for applications of these two most often-used empirical hazard function formulas in data analysis practice.


  1. 1.
    Ramakumar R (1993) Engineering reliability: fundamentals and applications. Prentice Hall, New JerseyGoogle Scholar
  2. 2.
    Meeker WQ, Escobar LA (1998) Statistical method for reliability data. Wiley, New JerseyGoogle Scholar
  3. 3.
    Davison AC (2003) Statistical models. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  4. 4.
    Venables WN, Ripley BD (1996) Modern applied statistics with S-Plus (corrected fourth printing). Springer, New YorkGoogle Scholar
  5. 5.
    Team RDC (2012) R: a language and environment for statistical computing. R Foundation for Statistical Computing, ViennaGoogle Scholar
  6. 6.
    Elsayed EA (1996) Reliability engineering. Addison Wesley Longman, Inc, ReadingGoogle Scholar
  7. 7.
    Rai B, Singh N (2003) Hazard rate estimation from incomplete and unclean warranty data. Reliab Eng Syst Saf 81:79–92CrossRefGoogle Scholar
  8. 8.
    Chung KL, AitSahlia F (2003) Elementary probability theory with stochastic process and an introduction to mathematical finance, 4th edn. Springer, New YorkGoogle Scholar
  9. 9.
    Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman & Hall/CRC, Boca RatonCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Chemistry, Physics and Mechanical EngineeringQueensland University of TechnologyBrisbaneAustralia

Personalised recommendations