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The exponentiation formula of reliability and survival: Does it always hold?

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Abstract

In reliability and biometry, it is common practice to choose a failure model by first assessing the failure rate function subjectively, and then invoking the well known “exponentiation formula”. The derivation of this formula is based on the assumption that the underlying failure distribution be absolutely continuous. Thus, implicit in the above approach is the understanding that the selected failure distribution will be absolutely continuous. The purpose of this note is to point out that the absolute continuity may fail when the failure rate is assessed conditionally, and in particular when it is conditioned on certain types of covariates, called “internal covariates”. When such is the case, the exponentiation formula should not be used.

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References

  • R. E. Barlow and F. Proschan,Statistical Theory of Reliability and Life Testing, 2nd ed. Silver Spring, MD: To Begin With, 1981.

    Google Scholar 

  • D. R. Cox, “Regression Models and Life Tables (with Discussion),”J. R. Statist. Soc., Ser. B vol. 39 pp. 86–94. 1972.

    Google Scholar 

  • T. R. Fleming and D. P. Harrington,Counting Processes and Survival Analysis. New York: John Wiley, 1991.

    Google Scholar 

  • J. D. Kalbfleisch and R. L. Prentice,The Statistical Analysis of Failure Time Data, New York: John Wiley, 1980.

    Google Scholar 

  • S. Lydersen,Reliability Testing Based on Deterioration Measurements, Doctoral Dissertation. University of Trondheim, Norway, 1988.

    Google Scholar 

  • R. L. Prentice, J. D. Kalbfleisch, A. V. Peterson Jr, N. Flournoy, V. T. Farewell, and N. E. Breslow, “The Analysis of Failure Times in the Presence of Competing Risks,”Biometrics vol. 34 pp. 541–54, 1979.

    Google Scholar 

  • N. D. Singpurwalla, “Survival in Dynamic Environments,”Statistical Science vol. 10 no. 1, to appear, 1995.

  • N. D. Singpurwalla and M. Youngreen, “Multivariate Distributions Induced by Dynamic Environments,”Scandinavian Journal of Statistics vol. 20 pp. 251–261, 1993.

    Google Scholar 

  • S. M. Wiederhorn and E. R. Fuller Jr., “Structural Reliability of Ceramic Materials,”Mat. Sci. Eng. vol. 71 pp. 169–86, 1985.

    Google Scholar 

  • A. Yashin and E. Arjas, “A Note on Random Intensities and Conditional Survival Functions,”J. Appl. Prob. vol. 25 pp. 630–635, 1988.

    Google Scholar 

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Authors and Affiliations

  1. The George Washington University, 20052, Washington D.C.

    Nozer D. Singpurwalla

  2. Trinity College, University of Dublin, Dublin, Ireland

    Simon P. Wilson

Authors
  1. Nozer D. Singpurwalla
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  2. Simon P. Wilson
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Singpurwalla, N.D., Wilson, S.P. The exponentiation formula of reliability and survival: Does it always hold?. Lifetime Data Anal 1, 187–194 (1995). https://doi.org/10.1007/BF00985769

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  • DOI: https://doi.org/10.1007/BF00985769

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