Bathtub-type curves in reliability and beyond

Australian and New Zealand Journal of Statistics

Volumn 49, Issue 3, 2007, Pages 251-265

  Bebbington, Mark ^a   Lai, Chin Diew ^a   Zitikis, Ričardas ^b  

^a MASSEY UNIVERSITY   (New Zealand)

^b UNIVERSITY OF WESTERN ONTARIO   (Canada)

Author keywords

Bathtub distribution; Burn in; Failure rate; Hazard rate; Infant mortality; Laplace transform; Mean residual life; Turning point; Useful period; Wear out phase; Weibull distribution

Indexed keywords

EID: 34548264625     PISSN: 13691473     EISSN: 1467842X     Source Type: Journal    
DOI: 10.1111/j.1467-842X.2007.00479.x     Document Type: Article

References (54)

1. Alzaid, A., Kim, J.S. & Proschan, F. (1991). Laplace ordering and its applications. J. Appl. Probab. 28, 116 130.
2. Bebbington, M., Lai, C.D. & Zitikis, R. (2005). Estimating the turning points of failure rate and mean residual life functions. Commun. Dependability Qual. Manag. 8, 11 15.
3. Bebbington, M., Lai, C.D. & Zitikis, R. (2006a). Useful periods for lifetime distributions with bathtub shaped hazard rate functions. IEEE Trans. Reliab. 55, 245 251.
4. Bebbington, M., Lai, C.D. & Zitikis, R. (2006b). Changing phases of human life expectancy: can we find them mathematically In Proceedings of the UNU-WIDER Conference on Advancing Health Equity, United Nations University-World Institute for Development Economics Research (UNU-WIDER), Helsinki, September 2006.
5. Bebbington, M., Lai, C.D. & Zitikis, R. (2007a). A flexible Weibull extension. Reliab. Eng. Syst. Safety 92, 719 726.
6. Bebbington, M., Lai, C.D. & Zitikis, R. (2007b). Optimum burn-in time for a bathtub shaped failure distribution. Methodol. Comput. Appl. Probab. 9, 1 20.
7. Bebbington, M., Lai, C.D. & Zitikis, R. (2007c). Modeling human mortality using mixtures of bathtub shaped failure distributions. J. Theor. Biol. 245, 528 538.
8. Bebbington, M., Lai, C.D. & Zitikis, R. (2007d). Estimating the turning point of a bathtub shaped failure distribution. J. Stat. Plan. Infer. (forthcoming).
9. Bebbington, M., Lai, C.D. & Zitikis, R. (2007e). Reduction in mean residual life in the presence of a constant competing risk. Appl. Stoch. Mod. Bus. (forthcoming).
10. Bekker, L. & Mi, J. (2003). Shape and crossing properties of mean residual life functions. Stat. Probab. Lett. 64, 225 234.
11. Belzunce, F., Ortega, E.-M. & Ruiz, J.M. (2007). On non-monotonic ageing properties from the Laplace transform, with actuarial applications. Insur. Math. Econ. 40, 1 14.
12. Block, H.W., Borges, W.S. & Savits, T.H. (1985). Age-dependent minimal repair. J. Appl. Probab. 22, 370 385.
13. Block, H.W., Jong, Y.-K. & Savits, T.H. (1999). Bathtub functions and burn-in. Probab. Eng. Inform. Sci. 13, 497 507.
14. Block, H.W., Li, Y. & Savits, T.H. (2005). Mixtures of normal distributions: Modality and failure rate. Stat. Probab. Lett. 74, 253 264.
15. Block, H.W. & Savits, T.H. (1997). Burn-in. Stat. Sci. 12, 1 19.
16. Block, H.W., Savits, T.H. & Singh, H. (2002). A criterion for burn-in that balances mean residual life and residual variance. Oper. Res. 50, 290 296.
17. Block, H.W., Savits, T.H. & Wondmagegnehu, E.T. (2003). Mixtures of distributions with increasing linear failure rates. J. Appl. Probab. 40, 485 504.
18. Brown, M. & Proschan, F. (1983). Imperfect repair. J. Appl. Probab. 20, 851 859.
19. Chang, D.S. (2000). Optimal burn-in decision for products with an unimodal failure rate function. Eur. J. Oper. Res. 126, 534 540.
20. Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Stat. Probab. Lett. 49, 155 161.
21. Csörgö, M. & Zitikis, R. (1996). Mean residual life processes. Ann. Stat. 24, 1717 1739.
22. Denuit, M. (2001). Laplace transform ordering of actuarial quantities. Insur. Math. Econ. 29, 83 102.
23. Ghai, G.L. & Mi, J. (1999). Mean residual life and its association with failure rate. IEEE Trans. Reliab. 48, 262 266.
24. Glaser, R.E. (1980). Bathtub and related failure rate characterizations. J. Am. Stat. Assoc. 75, 667 672.
25. Guess, F., Nam, K.H. & Park, D.H. (1998). Failure rate and mean residual life with trend changes. Asia-Pac. J. Oper. Res. 15, 239 244.
26. Gupta, R.C. & Akman, O.H. (1995). Mean residual life function for certain types of non-monotonic ageing. Commun. Stat. - Stoch. Mod. 11, 219 225.
27. Haines, A.L. & Singpurwalla, N.D. (1974). Some contributions to the stochastic characterization of wear. In Reliability and Biometry, eds F. Proshan & R. Serfling, pp. 47 80. Philadelphia : Society for Industrial and Applied Mathematics.
28. Hairer, E. & Wanner, G. 1996. Analysis by Its History. New York : Springer.
29. Hjorth, U. (1980). A reliability distribution with increasing, decreasing, constant and bathtub-shaped failure rates. Technometrics 22, 99 107.
30. Jensen, F. & Petersen, N.E. 1982. Burn-in: An Engineering Approach to the Design and Analysis of Burn-in Procedures. New York : Wiley.
31. Jiang, R. & Murthy, D.N.P. (1998). Mixture of Weibull distributions - Parametric characterization of failure rate function. Appl. Stoch. Mod. Data Anal. 14, 47 65.
32. Kececioglu, D. & Sun, F.-B. 1997. Burn-In Testing: Its Quantification and Optimization. Englewood Cliffs, NJ : Prentice Hall.
33. Klutke, G.-A., Kiessler, P.C. & Wortman, M.A. (2003). A critical look at the bathtub curve. IEEE Trans. Reliab. 52, 125 129.
34. Kuo, W. (1984). Reliability enhancement through optimal burn-in. IEEE Trans. Reliab. 33, 145 156.
35. Kuo, W. & Kuo, Y. (1983). Facing the headaches of early failures: A state-of-the-art review of burn-in decisions. Proc. IEEE 71, 1257 1266.
36. Lai, C.D. & Xie, M. 2006. Stochastic Ageing and Dependence for Reliability. New York : Springer.
37. Lai, C.D., Xie, M. & Murthy, D.N.P. (2001). Bathtub-shaped failure rate life distributions. In Handbook of Statistics, eds N. Balakrishnan & C.R. Rao, Vol. 20, Advances in Reliability, pp. 69 104, Amsterdam : Elsevier.
38. Lai, C.D., Xie, M. & Murthy, D.N.P. (2003). Modified Weibull model. IEEE Trans. Reliab. 52, 33 37.
39. Launer, R.L. (1993). Graphical techniques for analyzing failure data with the percentile residual-life function. IEEE Trans. Reliab. 42, 71-75, 80.
40. Lawrence, M.J. (1966). An investigation of the burn-in problem. Technometrics 8, 61 71.
41. Leemis, L.M. & Beneke, M. (1990). Burn-in models and methods: A review. IIE Trans. 22, 172 180.
42. Mi, J. (1994). Maximization of a survival probability and its application. J. Appl. Probab. 31, 1026 1033.
43. Mi, J. (1995). Bathtub failure rate and upside-down bathtub mean residual life. IEEE Trans. Reliab. 44, 388 391.
44. Mi, J. (1996). Minimizing some cost functions related to both burn-in and field use. Oper. Res. 44, 497 500.
45. Mi, J. (2004). A general approach to the shape of failure rate and MRL functions. Nav. Res. Logist. 51, 543 556.
46. Park, K.S. (1985). Effect of burn-in on mean residual life. IEEE Trans. Reliab. 34, 522 523.
47. Rajarshi, S. & Rajarshi, M.B. (1988). Bathtub distributions: A review. Commun. Stat. - Theor. Meth. 17, 2597 2621.
48. Savits, T.H. (2003). Preservation of generalized bathtub-shaped functions. J. Appl. Probab. 40, 473 484.
49. Smith, R.M. & Bain, L.J. (1975). An exponential power life-testing distribution. Commun. Stat. - Theor. Meth. 4, 469 481.
50. Wang, F.K. (2000). A new model with bathtub-shaped failure rate using an additive Burr XII distribution. Reliab. Eng. Syst. Safety 70, 305 312.
51. Watson, G.S. & Wells, W.T. (1961). On the possibility of improving the mean useful life of items by eliminating those with short lives. Technometrics 3, 281 298.
52. Wong, K.L. (1988). The bathtub does not hold water any more. Qual. Reliab. Eng. Int. 4, 279 282.
53. Xie, M. & Lai, C.D. (1995). Reliability analysis using additive Weibull model with bathtub-shaped failure rate function. Reliab. Eng. Syst. Safety 52, 87 93.
54. Xie, M., Tang, Y. & Goh, T.N. (2002). A modified Weibull extension with bathtub-shaped failure rate function. Reliab. Eng. Syst. Safety 76, 279 285.