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Multivariate Distributions in Reliability Theory and Life Testing

  • Henry W. Block
  • Thomas H. Savits
Part of the NATO Advanced study Institutes Series book series (ASIC, volume 79)

Summary

Multivariate parametric distributions which are of interest in reliability theory and life testing are discussed. These include distributions with exponential, Weibull and gamma univariate marginal distributions. Other distributions of interest are the multivariate nonparametric distributions whose marginals have increasing failure rates (IFR), increasing, failure rate averages (IFRA), are new better than used (NBU) or new better than used in expectation (NBUE). Also mentioned are univariate and multivariate processes which have associated with them distributions in the various nonparametric classes mentioned above.

Key Words

multivariate exponential distributions multivariate Weibull distributions multivariate gamma distributions multi-variate exponential extensions shock models threshold model gestation model characteristic function equation multivariate IFR multivariate IFRA multivariate NBU multivariate NBUE 

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References

  1. Arjas, E. (1979). A stochastic process approach to multivariate reliability systems. Unpublished report.Google Scholar
  2. Arnold, B.C. (1967). A note on multivariate distributions with specified marginals. Journal of American Statistical Association, 62, 1460–1461.CrossRefGoogle Scholar
  3. Arnold, B.C. (1975a). A characterization of the exponential distribution by multivariate geometric compounding. Sankhya, Series A, 37, 164–173.zbMATHGoogle Scholar
  4. Arnold, B. C. (1975b). Multivariate exponential distributions based on heirarchical successive damage. Journal of Applied Probability, 12, 142–147.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Bain, L.J. (1978). Statistical Analysis of Reliability and Life-Testing Models. Marcel Dekker, New York.zbMATHGoogle Scholar
  6. Barlow, R.E., Proschan, F. (1965). Mathematical Theory of Reliability. Wiley, New York.zbMATHGoogle Scholar
  7. Barlow, R.E., Proschan, F. (1975). Statistical Theory of Reliability and Life-Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
  8. Basu, A.P., Block, H. (1975). On characterizing univariate and multivariate exponential distributions with applications. In Statistical Distributions in Scientific Work, Vol. 3, G.P. Patil, S. Kotz, J. K. Ord, eds. Reidel, Dordrecht- Holland.Google Scholar
  9. Bemis, B.M., Bains, L.J., Higgins, J.J. (1972). Estimation and hypothesis testing for the parameters of a bivariate exponential distribution. Journal of American Statistical Association, 67, 927–929.zbMATHCrossRefGoogle Scholar
  10. Bhattacharya, G.K., Johnson, R.A. (1973). On a test of independence in a bivariate exponential distribution. Journal of American Statistical Association, 68, 704–706.CrossRefGoogle Scholar
  11. Block, H.W. (1975). Continuous multivariate exponential extensions. In Reliability and Fault Tree Analysis, R.E. Barlow, J.B. Fussell, N.D. Singpurwalla, eds. SIAM, Philadelphia.Google Scholar
  12. Block, H.W. (1977a). A characterization of a bivariate exponential distribution. Annals of Statistics, 5, 808–812.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Block, H.W. (1977b). A family of bivariate life distributions. In The Theory and Applications of Reliability, Vol. I, C. P. Tsokos, I. Shimi, eds. Academic Press, New York.Google Scholar
  14. Block, H.W., Basu, A.P. (1974). A continuous bivariate exponential extension. Journal of American Statistical Association, 69, 1031–1037.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Block H.W., Paulson, A.S., Kohberger (1976). Some bivariate exponential distributions: syntheses and properties. Unpublished report.Google Scholar
  16. Block, H.W., Savits, T.H. (1976). The IFRA closure problem. Annals of Probability, 4, 1030–1032.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Block, H.W., Savits, T.H. (1978a). Shock models with NBUE survival. Journal of Applied Probability, 15, 621–628.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Block, H.W., Savits, T.H. (1978b). The class of MIFRA lifetimes and its relation to other classes. University of Pittsburgh Research Report #78-01.Google Scholar
  19. Block, H.W., Savits, T.H. (1979a). Laplace transforms for classes of life distributions. Annals of Probability, to appear.Google Scholar
  20. Block, H.W., Savits, T.H. (1979b). Multivariate IFRA distributions. Annals of Probability, to appear.Google Scholar
  21. Block, H.W., Savits, T.H. (1979c). Multivariate IFRA processes. Annals of Probability, to appear.Google Scholar
  22. Block, H.W., Savits, T.H. (1980). Multivariate classes in reliability theory. Mathematics of Operations Research, to appear.Google Scholar
  23. Bryant, J.L. (1979). Stochastic properties of cycling systems with random lifetimes. Unpublished report.Google Scholar
  24. Buchanan, W.B., Singpurwalla, N.D. (1977). Some stochastic characterizations of multivariate survival. In The Theory and Applications of Reliability, Vol. I, C. P. Tsokos, I Shimi, eds. Academic Press, New York.Google Scholar
  25. David, H.A. (1974). Parametric approaches to the theory of competing risks. In Reliability and Biometry, F. Proschan, R.J. Serfling, eds. SIAM, Philadelphia.Google Scholar
  26. Downton, F. (1970). Bivariate exponential distributions in reliability theory. Journal of the Royal Statistical Society, Series B, 32, 408–417.MathSciNetzbMATHGoogle Scholar
  27. El-Neweihi, E., Proschan, F., Sethuraman, J. (1978). Multi- state coherent systems. Journal of Applied Probability, 15, 675–688.MathSciNetzbMATHCrossRefGoogle Scholar
  28. El-Neweihi, E., Proschan, F., Sethuraman, J. (1980). A multivariate new better than used class derived from a shock model FSU Statistics Report M540.Google Scholar
  29. Esary, J.D., Marshall, A.W. (1973). Multivariate geometric distributions generated by a cumulative damage process. Naval Postgraduate School Report NP555EY73041A.Google Scholar
  30. Esary, J.D. and Marshall, A. W. (1974). Multivariate distri-butions with exponential minimums. Annals of Statistics, 2, 84–96.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Esary, J.D., Marshall, A.W. (1979). Multivariate distributions with increasing hazard rate average. Annals of Probability, 7, 359–370.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Freund, J. (1961). A bivariate extension of the exponential distribution. Journal of American Statistical Association, 56, 971–977.MathSciNetzbMATHCrossRefGoogle Scholar
  33. Friday, D.S., Patil, G.P. (1977). A bivariate exponential model with applications to reliability and computer generation of random variables. In Theory and Applications of Reliability, Vol. I, C. P. Tsokos, I. Shimi, eds. Academic Press, New York.Google Scholar
  34. Galambos, J., Kotz, S. (1978). Characterizations of Probability Distributions. Springer-Verlag, Berlin.zbMATHGoogle Scholar
  35. Ghosh, M., Ebrahimi, N. (1980). Multivariate NBU and NBUE Distributions. Unpublished report.Google Scholar
  36. Griffith, W. (1980). Multistate reliability models. Journal of Applied Probability, to appear.Google Scholar
  37. Gross, A.J. (1973). A competing risk model: a one organ subsystem plus a two organ subsysytem. IEEE Transactions on Reliability, R-22, 24–27.Google Scholar
  38. Gross, A.J., Clark, V.A. and Liu, V. (1971). Estimation of survival parameters when one of two organs must function for survival. Biometrics, 27, 369–377.CrossRefGoogle Scholar
  39. Gross, A.J., Lam, C.F. (1979). Paired observations from a survival distribution. Unpublished report.Google Scholar
  40. Hawkes, A.G. (1972). A bivariate exponential distribution with applications to reliability. Journal of the Royal Statistical Society, Series B, 34, 129–131.zbMATHGoogle Scholar
  41. Hsu, C.L., Shaw, L., Tyan, S.G. (1977). Reliability applications of multivariate exponential distributions. Polytechnic Institute of New York Report POLY-EE-77-036.Google Scholar
  42. Johnson, N.L., Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.zbMATHGoogle Scholar
  43. Krishnaiah, P.R. (1977). On generalized multivariate gamma type distributions and their applications in reliability. In Theory and Applications of Reliability, Vol. I, C.P. Tsokos, I. Shimi, eds. Academic Press, New York.Google Scholar
  44. Krishnaiah, P.R., Rao, M.M. (1961). Remarks on a multivariate gamma distribution. American Mathematical Monthly, 68, 342–346.MathSciNetzbMATHCrossRefGoogle Scholar
  45. Krishnamoorthy, A.S., Parthasarathy, M. (1951). A multivariate gamma type distribution. Annals of Mathematical Statistics, 22, 549–557.MathSciNetzbMATHCrossRefGoogle Scholar
  46. Lee, L., Thompson, Jr., W.A. (1974). Results on failure time and pattern for the series system. In Reliability and Biometry. F. Proschan and R. J. Serfling, eds. SIAM, Philadelphia.Google Scholar
  47. Mann, N.R., Schafer, R. E., Singpurwalla, N. D. (1974). Methods for Statistical Analysis of Reliability and Life Data. Wiley, New York.zbMATHGoogle Scholar
  48. Marshall, A. W. (1975). Multivariate distributions with monotone hazard rate. In Reliability and Fault Tree Analysis, R.E. Barlow, J. Fussell, N.D. Singpurwalla, eds. SIAM, Philadelphia.Google Scholar
  49. Marshall, A.W., Olkin, I. (1967a). A multivariate exponential distribution. Journal of American Statistical Association, 62, 30–44.MathSciNetzbMATHCrossRefGoogle Scholar
  50. Marshall, A.W., Olkin, I. (1967b). A generalized bivariate exponential distribution. Journal of Applied Probability, 4, 291–302.MathSciNetzbMATHCrossRefGoogle Scholar
  51. Marshall, A.W., Shaked, M. (1979). A class of multivariate new better than used distributions. Unpublished report.Google Scholar
  52. Mehrotra, K.G., Michalek, J.E. (1976). Estimation of parameters and tests of independence in a continuous bivariate exponential distribution. Unpublished manuscript.Google Scholar
  53. Moeschberger, M. L. (1974). Life tests under dependent competing causes of failure. Technometrics, 16, 39–47.MathSciNetzbMATHCrossRefGoogle Scholar
  54. Paulson, A.S. (1973). A characterization of the exponential distribution and a bivariate exponential distribution. Sankhya, Series A, 35, 69–78.MathSciNetzbMATHGoogle Scholar
  55. Paulson, A.S., Uppuluri, V.R.R. (1972a). Limit laws of a sequence determined by a random difference equation governing a one compartment system. Mathematical Biosciences, 13, 325–333.MathSciNetzbMATHCrossRefGoogle Scholar
  56. Paulson, A. S., Uppuluri, V. R. R. (1972b). A characterization of the geometric distribution and a bivariate - geometric distribution. Sankhya, Series A, 34, 88–91.MathSciNetGoogle Scholar
  57. Proschan, F., Sullo, P. (1974). Estimating the parameters of a bivariate exponential distribution in several sampling situations. In Reliability and Biometry, F. Proschan, R.J. Serfling, eds. SIAM, Philadelphia.Google Scholar
  58. Proschan, F., Sullo, P. (1976). Estimating the parameters of a multivariate exponential distribution. Journal of American Statistical Association, 71, 465–472.MathSciNetzbMATHCrossRefGoogle Scholar
  59. Ross, S. (1979). Multivalued state component systems. Annals of Probability, 7. 379–383.MathSciNetzbMATHCrossRefGoogle Scholar
  60. Spurrier, J. D. and Weier, D. R. (1979). A bivariate survival model derived from a Weibull distribution. University of South Carolina Statistics Technical Report No. 46.Google Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1981

Authors and Affiliations

  • Henry W. Block
    • 1
    • 2
  • Thomas H. Savits
    • 2
  1. 1.University of São PauloBrazil
  2. 2.University of PittsburghUSA

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