Advertisement

Stochastic Orders in Reliability

  • N. Unnikrishnan Nair
  • P. G. Sankaran
  • N. Balakrishnan
Chapter
Part of the Statistics for Industry and Technology book series (SIT)

Abstract

Stochastic orders enable global comparison of two distributions in terms of their characteristics. Specifically, for a given characteristic A, stochastic order says that the distribution of X has lesser (greater) A than the distribution of Y. For example, one may use hazard rate or mean residual life for such a comparison. In this chapter, we discuss various stochastic orders useful in reliability modelling and analysis.The stochastic order treated here are the usual stochastic order, hazard rate order, mean residual life order, harmonic mean residual life order, renewal and harmonic renewal mean residual life orders, variance residual life order, percentile residual life order, reversed hazard rate order, mean inactivity time order, variance inactivity time order, the total time on test transform order, the convex transform (IHR) order, star (IHRA) order, DMRL order, superadditive (NBU) order, NBUE order, NBUHR and NBUHRA orders and MTTF order. The interpretation of ageing concepts, preservation properties with reference to convolution, mixing and coherent structures are also discussed in relation to each of these orders. Implications among the different orders are also presented. Examples of the stochastic orders and counter examples where certain implications do not hold are also provided. Some special models used in reliability like proportional hazard and reverse hazard models, mean residual life models and weighted distributions have been discussed in earlier chapters. Some applications of these stochastic models are reviewed as well.

Keywords

Hazard Rate Residual Life Quantile Function Stochastic Order Life Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 11.
    Aboukalam, F., Kayid, M.: Some new results about shifted hazard and shifted likelihood ratio orders. Int. Math. Forum 31, 1525–1536 (2007)MathSciNetGoogle Scholar
  2. 13.
    Abraham, B., Nair, N.U.: A criterion to distinguish ageing patterns. Statistics 47, 85–92 (2013)MathSciNetMATHCrossRefGoogle Scholar
  3. 23.
    Ahmad, I.A., Kayid, M.: Characterization of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions. Probab. Eng. Inform. Sci. 19, 447–461 (2005)MathSciNetGoogle Scholar
  4. 24.
    Ahmad, I.A., Kayid, M., Pellerey, F.: Further results involving the MIT order and the IMIT class. Probab. Eng. Inform. Sci. 19, 377–395 (2005)MathSciNetMATHGoogle Scholar
  5. 28.
    Ahmed, A.N.: Preservation properties of the mean residual life ordering. Stat. Paper. 29, 143–150 (1988)MATHCrossRefGoogle Scholar
  6. 35.
    Alzaid, A.A.: Mean residual life ordering. Stat. Paper. 29, 35–43 (1988)MathSciNetMATHCrossRefGoogle Scholar
  7. 39.
    Asha, G., Nair, N.U.: Reliability properties of mean time to failure in age replacement models. Int. J. Reliab. Qual. Saf. Eng. 17, 15–26 (2010)CrossRefGoogle Scholar
  8. 68.
    Barlow, R.E., Proschan, F.: Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York (1975)MATHGoogle Scholar
  9. 78.
    Bartoszewicz, J., Skolimowska, M.: Preservation of stochastic orders under mixtures of exponential distributions. Probab. Eng. Inform. Sci. 20, 655–666 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 86.
    Belzunce, F., Navarro, J., Ruiz, J.M., Aguila, Y.D.: Some results on residual entropy function. Metrika 59, 147–161 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 101.
    Birnbaum, Z.W.: On random variables with comparable peakedness. Ann. Math. Stat. 19, 76–81 (1948)MathSciNetMATHCrossRefGoogle Scholar
  12. 111.
    Block, H.W., Savits, T.H., Singh, H.: The reversed hazard rate function. Probab. Eng. Inform. Sci. 12, 69–90 (1998)MathSciNetMATHCrossRefGoogle Scholar
  13. 114.
    Boland, P.J., Shaked, M., Shanthikumar, J.G.: Stochastic ordering of order statistics. In: Balakrishnan, N., Rao, C.R. (eds.) Handbook of Statistics 16, Order Statistics: Theory and Methods, pp. 89–103. North-Holland, Amsterdam (1998)Google Scholar
  14. 115.
    Boland, P.J., Singh, H., Cukic, B.: Stochastic orders in partition and random testing of software. J. Appl. Probab. 39, 555–565 (1999)MathSciNetGoogle Scholar
  15. 117.
    Bon, J., Paltanea, E.: Ordering properties of convolution of exponential random variables. Lifetime Data Anal. 5, 185–192 (1999)MathSciNetMATHCrossRefGoogle Scholar
  16. 120.
    Brito, G., Zequeira, R.I., Valdes, J.G.: On hazard and reversed hazard rate orderings in two component series systems with active redundancies. Stat. Probab. Lett. 81, 201–206 (2011)MathSciNetMATHCrossRefGoogle Scholar
  17. 164.
    Da, G., Ding, W., Li, X.: On hazard rate ordering of parallel systems with two independent components. J. Stat. Plann. Infer. 140, 2148–2154 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 173.
    Deshpande, J.V., Singh, H., Bagai, I., Jain, K.: Some partial orders describing positive ageing. Comm. Stat. Stoch. Model. 6, 471–481 (1990)MathSciNetMATHCrossRefGoogle Scholar
  19. 177.
    DiCresenzo, A.: Some results on the proportional reversed hazards model. Stat. Probab. Lett. 50, 313–321 (2000)CrossRefGoogle Scholar
  20. 190.
    Fagiouli, E., Pellerey, F.: Mean residual life and increasing convex comparison of shock models. Stat. Probab. Lett. 20, 337–345 (1993)CrossRefGoogle Scholar
  21. 192.
    Fagiouli, E., Pellerey, F.: Moment inequalities for sums of DMRL random variables. J. Appl. Probab. 34, 525–535 (1997)MathSciNetCrossRefGoogle Scholar
  22. 202.
    Franco-Pereira, A.M., Lillo, R.E., Romo, J., Shaked, M.: Percentile residual life orders. Appl. Stoch. Model. Bus. Ind. 27, 235–252 (2011)MathSciNetCrossRefGoogle Scholar
  23. 239.
    Gupta, R.C., Gupta, P.L., Gupta, R.D.: Modelling failure time data with Lehmann alternative. Comm. Stat. Theor. Meth. 27, 887–904 (1998)MATHCrossRefGoogle Scholar
  24. 241.
    Gupta, R.C., Kirmani, S.N.U.A.: On order relationships between reliability measures. Comm. Stat. Stoch. Model. 3, 149–156 (1987)MathSciNetMATHCrossRefGoogle Scholar
  25. 254.
    Gupta, R.D., Nanda, A.K.: Some results on reversed hazard rate ordering. Comm. Stat. Theor. Meth. 30, 2447–2457 (2001)MathSciNetMATHCrossRefGoogle Scholar
  26. 284.
    Hu, C., Lin, G.D.: Characterization of the exponential distribution by stochastic ordering properties of the geometric compound. Ann. Inst. Stat. Math. 55, 499–506 (2003)MathSciNetMATHCrossRefGoogle Scholar
  27. 285.
    Hu, T., He, P.: A note on comparisons of k-out-of n systems with respect to hazard and reversed hazard rate orders. Probab. Eng. Inform. Sci. 14, 27–32 (2000)MathSciNetMATHGoogle Scholar
  28. 286.
    Hu, T., Wei, Y.: Stochastic comparison of spacings from restricted families of distributions. Stat. Probab. Lett. 53, 91–99 (2001)MathSciNetMATHCrossRefGoogle Scholar
  29. 288.
    Hu, T., Zhu, Z., Wei, Y.: Likelihood ratio and mean residual life orders for order statistics of heterogeneous random variables. Probab. Eng. Inform. Sci. 15, 259–272 (2001)MathSciNetMATHCrossRefGoogle Scholar
  30. 298.
    Joag-Dev, K., Kochar, S.C., Proschan, F.: A general comparison theorem and its applications to certain partial orderings of distributions. Stat. Probab. Lett. 22, 111–119 (1995)MathSciNetMATHCrossRefGoogle Scholar
  31. 301.
    Joe, H., Proschan, F.: Percentile residual life functions. Oper. Res. 32, 668–678 (1984)MathSciNetMATHCrossRefGoogle Scholar
  32. 319.
    Kayid, M., Ahmad, I.A.: On the mean inactivity time ordering with reliability applications. Probab. Eng. Inform. Sci. 18, 395–409 (2004)MathSciNetMATHCrossRefGoogle Scholar
  33. 320.
    Kayid, M., El-Bassiouny, A.H., Al-Wasel, I.A.: On some new stochastic orders of interest in reliability theory. Int. J. Reliab. Appl. 8, 95–109 (2007)Google Scholar
  34. 321.
    Kebir, Y.: Laplace transform characterizations of probabilistic orderings. Probab. Eng. Inform. Sci. 8, 125–134 (1994)CrossRefGoogle Scholar
  35. 325.
    Kijima, M.: Hazard rate and reversed hazard rate monotonicities in continuous Markov chains. J. Appl. Probab. 35, 545–556 (1998)MathSciNetMATHCrossRefGoogle Scholar
  36. 328.
    Kirmani, S.N.U.A.: On sample spacings from IMRL distributions. Stat. Probab. Lett. 29, 159–166 (1996)MathSciNetMATHCrossRefGoogle Scholar
  37. 329.
    Kirmani, S.N.U.A.: On sample spacings from IMRL distributions. Stat. Probab. Lett. 37, 315 (1998)MathSciNetMATHCrossRefGoogle Scholar
  38. 346.
    Kochar, S.C.: Distribution free comparison of two probability distributions with reference to their hazard rates. Biometrika 66, 437–441 (1979)MathSciNetMATHCrossRefGoogle Scholar
  39. 347.
    Kochar, S.C.: On extensions of DMRL and related partial orderings of life distributions. Comm. Stat. Stoch. Model. 5, 235–245 (1989)MathSciNetMATHCrossRefGoogle Scholar
  40. 349.
    Kochar, S.C., Li, X., Shaked, M.: The total time on test transform and the excess wealth stochastic orders of distributions. Adv. Appl. Probab. 34, 826–845 (2002)MathSciNetMATHCrossRefGoogle Scholar
  41. 350.
    Kochar, S.C., Wiens, D.: Partial orderings of life distributions with respect to their ageing properties. Nav. Res. Logist. 34, 823–829 (1987)MathSciNetMATHCrossRefGoogle Scholar
  42. 352.
    Korwar, R.: On stochastic orders for the lifetime of k-out-of-n system. Probab. Eng. Inform. Sci. 17, 137–142 (2003)MathSciNetMATHCrossRefGoogle Scholar
  43. 381.
    Lefevre, C., Utev, S.: Comparison of individual risk models. Insur. Math. Econ. 28, 21–30 (2001)MathSciNetMATHCrossRefGoogle Scholar
  44. 383.
    Lehmann, E.L., Rojo, J.: Invariant directional orderings. Ann. Stat. 20, 2100–2110 (1992)MathSciNetMATHCrossRefGoogle Scholar
  45. 388.
    Li, H., Shaked, M.: A general family of univariate stochastic orders. J. Stat. Plann. Infer. 137, 3601–3610 (2007)MathSciNetMATHCrossRefGoogle Scholar
  46. 392.
    Li, X., Shaked, M.: The observed total time on test and the observed excess wealth. Stat. Probab. Lett. 68, 247–258 (2004)MathSciNetMATHCrossRefGoogle Scholar
  47. 393.
    Li, X., Xu, M.: Reversed hazard rate order of equilibrium distributions and a related ageing notion. Stat. Paper. 49, 749–767 (2007)CrossRefGoogle Scholar
  48. 395.
    Li, X., Zuo, M.J.: Preservation of stochastic orders for random minima and maxima with applications. Nav. Res. Logist. 51, 332–344 (2000)MathSciNetCrossRefGoogle Scholar
  49. 406.
    Ma, C.: A note on stochastic ordering of order statistics. J. Appl. Probab. 34, 785–789 (1997)MathSciNetMATHCrossRefGoogle Scholar
  50. 408.
    Mahdy, M.: Characterization and preservations of the variance inactivity time ordering and increasing variance inactivity time class. J. Adv. Res. 3, 29–34 (2012)CrossRefGoogle Scholar
  51. 409.
    Mann, H.B., Whitney, D.R.: On a test of whether one of two random variables is stochastically larger than the other. Ann. Math. Stat. 18, 50–60 (1947)MathSciNetMATHCrossRefGoogle Scholar
  52. 412.
    Marshall, A.W., Olkin, I.: Life Distributions. Springer, New York (2007)MATHGoogle Scholar
  53. 417.
    Misra, N., Gupta, N., Dhariyal, I.D.: Preservation of some ageing properties and stochastic orders by weighted distributions. Comm. Stat. Theor. Meth. 37, 627–644 (2008)MathSciNetMATHCrossRefGoogle Scholar
  54. 432.
    Muller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002)Google Scholar
  55. 446.
    Nair, N.U., Sankaran, P.G.: Some results on an additive hazard model. Metrika 75, 389–402 (2010)MathSciNetCrossRefGoogle Scholar
  56. 447.
    Nair, N.U., Sankaran, P.G., Vineshkumar, B.: Total time on test transforms and their implications in reliability analysis. J. Appl. Probab. 45, 1126–1139 (2008)MathSciNetMATHCrossRefGoogle Scholar
  57. 458.
    Nanda, A.K., Bhattacharjee, S., Alam, S.S.: Properties of proportional mean residual life model. Stat. Probab. Lett 76, 880–890 (2006)MathSciNetMATHCrossRefGoogle Scholar
  58. 459.
    Nanda, A.K., Bhattacharjee, S., Balakrishnan, N.: Mean residual life function, associated orderings and properties. IEEE Trans. Reliab. 59, 55–65 (2010)CrossRefGoogle Scholar
  59. 460.
    Nanda, A.K., Jain, K., Singh, H.: On closure of some partial orderings under mixtures. J. Appl. Probab. 33, 698–706 (1996)MathSciNetMATHCrossRefGoogle Scholar
  60. 461.
    Nanda, A.K., Shaked, M.: The hazard rate and RHR orders with applications to order statistics. Ann. Inst. Stat. Math. 53, 853–864 (2001)MathSciNetMATHCrossRefGoogle Scholar
  61. 462.
    Nanda, A.K., Singh, H., Misra, N., Paul, P.: Reliability properties of reversed residual life time. Comm. Stat. Theor. Meth. 32, 2031–2042 (2003)MathSciNetMATHCrossRefGoogle Scholar
  62. 467.
    Navarro, J., Lai, C.D.: Ordering properties of systems with two dependent components. Comm. Stat. Theor. Meth. 36, 645–655 (2007)MathSciNetMATHCrossRefGoogle Scholar
  63. 474.
    Ortega, E.M.M.: A note on some functional relationships involving the mean inactivity time order. IEEE Trans. Reliab. 58, 172–178 (2009)CrossRefGoogle Scholar
  64. 490.
    Pellerey, F.: On the preservation of some orderings of risks under convolution. Insur. Math. Econ. 16, 23–30 (1995)MathSciNetMATHCrossRefGoogle Scholar
  65. 521.
    Scarsini, M., Shaked, M.: Some conditions for stochastic equality. Nav. Res. Logist. 37, 617–625 (1990)MathSciNetMATHCrossRefGoogle Scholar
  66. 526.
    Sengupta, D., Deshpande, J.V.: Some results on the relative ageing of two life distributions. J. Appl. Probab. 31, 991–1003 (1994)MathSciNetMATHCrossRefGoogle Scholar
  67. 531.
    Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, New York (2007)MATHCrossRefGoogle Scholar
  68. 533.
    Shaked, M., Wang, T.: Preservation of stochastic orderings under random mapping by point processes. Probab. Eng. Inform. Sci. 9, 563–580 (1995)CrossRefGoogle Scholar
  69. 541.
    Singh, H.: On partial orderings of life distributions. Nav. Res. Logist. 36, 103–110 (1989)MATHCrossRefGoogle Scholar
  70. 557.
    Szekli, R.: Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statistics, vol. 97. Springer, New York (1995)Google Scholar
  71. 597.
    Yu, Y.: Stochastic orderings of exponential family of distributions and their mixtures. J. Appl. Probab. 46, 244–254 (2009)MathSciNetMATHCrossRefGoogle Scholar
  72. 599.
    Zang, Z., Li, X.: Some new properties of stochastic ordering and ageing properties of coherent systems. IEEE Trans. Reliab. 59, 718–724 (2010)CrossRefGoogle Scholar
  73. 602.
    Zhao, P., Balakrishnan, N.: Mean residual life order of convolutions of heterogeneous exponential random variables. J. Multivariate Anal. 100, 1792–1801 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • N. Unnikrishnan Nair
    • 1
  • P. G. Sankaran
    • 1
  • N. Balakrishnan
    • 2
  1. 1.Department of StatisticsCochin University of Science and TechnologyCochinIndia
  2. 2.Department of Mathematics and StatisticsMcMasters UniversityHamiltonCanada

Personalised recommendations