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A new family of life distributions

  • Z.W. Birnbaum (a1) and S.C. Saunders (a2)
Summary

A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

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[1] Barlow, Richard E. and Proschan, Frank (1965) Mathematical Theory of Reliability. John Wiley and Sons, New York.
[2] Birnbaum, Z. W., Esary, J. D. and Marshall, A. W. (1966) Stochastic characterization of wear-out for components and systems. Ann. Math. Statist. 37, 816825.
[3] Birnbaum, Z. W. and Saunders, S. C. (1968) A probabilistic interpretation of Miner's rule. SIAM J. Appl. Math. 16, 637652.
[4] Birnbaum, Z. W. and Saunders, S. C. (1958) A statistical model for life length of material. J. Amer. Statist. Assoc. 53, 151160.
[5] CraméR, H. (1951) Mathematical Methods of Statistics. Princeton University Press.
[6] Feller, W. (1966) An Introduction to Probability Theory and its Applications II. John Wiley and Sons, New York.
[7] Freudenthal, A. M. and Shinozuka, M. (1961) Structural safety under conditions of ultimate load-failure and fatigue. WADD Technical Report 6177.
[8] Gumbel, E. J. (1958) Statistics of Extremes. Columbia University Press, New York.
[9] Parzen, E. (1967) On models for the probability of fatigue failure of a structure. Time Series Analysis Papers , 532548, Holden Day, San Francisco.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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