Weibull Distributions and Their Applications

Springer Handbooks

Volumn , Issue , 2006, Pages 63-78

  Lai, Chin Diew ^a   Murthy, D N ^b   Xie, Min ^c  

^a MASSEY UNIVERSITY   (New Zealand)

^b UNIVERSITY OF QUEENSLAND   (Australia)

^c NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Compete Risk Model; Failure Rate Function; Mean Residual Life; Weibull Distribution; Weibull Model

Indexed keywords

EID: 85088305870     PISSN: 25228692     EISSN: 25228706     Source Type: Book Series    
DOI: 10.1007/978-1-84628-288-1_3     Document Type: Chapter

References (81)

1. 3.1 A. J. Jr. Hallinan: A review of the Weibull distribution, J. Qual. Technol. 25, 85–93 (1993)
2. 3.2 N. L. Johnson, S. Kotz, N. Balakrishnan: Continuous Univariate Distributions, Vol. 1, 2nd edn. (Wiley, New York 1994)
3. 3.3 D. N. P. Murthy, M. Xie, R. Jiang: Weibull Models (Wiley, New York 2003)
4. 3.4 W. Weibull: A statistical theory of the strength of material, Ing. Vetenskapa Acad. Handlingar 151, 1–45 (1939)
5. 3.5 W. Weibull: A statistical distribution function of wide applicability, J. Appl. Mech. 18, 293–296 (1951)
6. 3.6 M. Fréchet: Sur la loi de probabilité de l’écart maximum, Ann. Soc. Polonaise Math. Cracovie 6, 93–116 (1927)
7. 3.7 R. A. Fisher, L. M. C. Tippett: Limiting forms of frequency distribution of the largest or smallest member of a sample, Proc. Cambridge Philos. Soc. 24, 180–190 (1928)
8. 3.8 N. L. Johnson, S. Kotz, N. Balakrishnan: Continuous Univariate Distributions, Vol. 2, 2nd edn. (Wiley, New York 1995)
9. 3.9 W. J. Zimmer, J. B. Keats, F. K. Wang: The Burr XII distribution in reliability analysis, J. Qual. Technol. 30(4), 386–394 (1998)
10. 3.10 C. D. Lai, M. Xie: Relative ageing for two parallel systems and related problems, Math. Comp. Model. 38, 1339–1345 (2003)
11. 3.11 M. Xie, C. D. Lai: On the increase of the expected lifetime by parallel redundancy, Asia Pacific J. Oper. Res. 13, 171–179 (1996)
12. 3.12 J. F. Lawless: Statistical Models and Methods for Lifetime Data (Wiley, New York 1982)
13. 3.13 W. Nelson: Applied Life Data Analysis (Wiley, New York 1982)
14. 3.14 M. Liao, T. Shimokawa: A new goodness-of-fit test for Type-I extreme-value, 2-parameter Weibull distributions with estimated parameters, J. Stat. Comput. Sim. 64, 23–48 (1999)
15. 3.15 W. Nelson, V. C. Thompson: Weibull probability papers, J. Qual. Technol. 3, 140–146 (1971)
16. 3.16 W. Nelson: Theory and application of hazard plotting for censored failure data, J. Qual. Technol. 14, 935– 966 (1972)
17. 3.17 J. D. Kalbfleisch, R. L. Prentice: The Statistical Analysis of Failure Data (Wiley, New York 1980)
18. 3.18 A. C. Cohen, B. J. Whitten, Y. Ding: Modified moment estimation for the three-parameter Weibull distribution, J. Qual. Technol. 16, 159–167 (1984)
19. 3.19 S. H. Zanakis, J. Kyparisis: A review of maximum likelihood estimation methods for the three-parameter Weibull distribution, J. Stat. Comp. Simul. 25, 53–73 (1986)
20. 3.20 B. Dodson: Weibull Analysis (ASQC Quality, Milwaukee 1994) see also 3.81
21. 3.21 M. Xie, C. D. Lai, D. N. P. Murthy: Weibull-related distributions for modelling of bathtub shaped failure rate functions. In: Mathematical and Statistical Methods in Reliability, ed. by B. H. Lindqvist, K. A. Doksum (World Scientific, Singapore 2003) pp. 283–297
22. 3.22 C. D. Lai, L. Y. Zhang, M. Xie: Mean residual life and other properties of Weibull related bathtub shape failure rate distributions, Int. J. Reliab. Qual. Saf. Eng. 11, 113–132 (2004)
23. 3.23 D. N. P. Murthy, J. Baik, M. Bulmer, R. J. Wilson: Two-dimensional modelling of failures. In: Springer Handbook Of Engineering Statistics, ed. by H. Pham (Springer, Heidelberg, Berlin 2004)
24. 3.24 A. W. Marshall, I. Olkin: A new method for adding aparameter to afamily of distributions with application to the exponential and Weibull families, Biometrika 84, 641–652 (1997)
25. 3.25 C. D. Lai, M. Xie, M. D. N. P. Murthy: Modified Weibull model, IEEE Trans. Reliab. 52, 33–37 (2003)
26. 3.26 G. S. Mudholkar, D. K. Srivastava: Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Trans. Reliab. 42, 299–302 (1993)
27. 3.27 R. Jiang, D. N. P. Murthy: The exponentiated Weibull family: A graphical approach, IEEE Trans. Reliab. 48, 68–72 (1999)
28. 3.28 J. A. Kies: The Strength of Glass, Report, Vol. 5093 (Naval Research Lab., Washington 1958)
29. 3.29 M. Xie, T. N. Goh, Y. Tang: A modified Weibull extension with bathtub-shaped failure rate function, Reliab. Eng. Syst. Saf. 76, 279–285 (2002)
30. 3.30 R. M. Smith, L. J. Bain: An exponential power life-testing distribution, Commun. Stat. 4, 469–481 (1975)
31. 3.31 Z. Chen: A new two-parameter distribution with bathtub shape or increasing failure rate function, Stat. Probab. Lett. 49, 155–161 (2000)
32. 3.32 R. Jiang, D. N. P. Murthy: Modeling failure data by mixture of two Weibull distributions: A graphical approach, IEEE Trans. Reliab. 44, 477–488 (1995)
33. 3.33 R. Jiang, D. N. P. Murthy: Mixture of Weibull distributions-parametric characterization of failure rate function, Stochastic Models Data Anal. 14, 47–65 (1998)
34. 3.34 J. Gurland, J. Sethuraman: How pooling failure data may reverse increasing failure rates, J. Am. Stat. Ass. 90, 1416–1423 (1995)
35. 3.35 R. Jiang, D. N. P. Murthy: Parametric study of competing risk model involving two Weibull distributions, Int. J. Reliab. Quality Saf. Eng. 4, 17–34 (1997)
36. 3.36 R. Jiang, D. N. P. Murthy: Parametric study of multiplicative model involving two Weibull distributions, Reliab. Eng. Sys. Saf. 55, 217–226 (1997)
37. 3.37 R. Jiang, D. N. P. Murthy: Reliability modeling involving two Weibull distributions, Reliab. Eng. Sys. Saf. 47, 187–198 (1995)
38. 3.38 R. Jiang, M. Zuo, D. N. P. Murthy: Two sectional models involving two Weibull distributions, Int. J. Reliab. Qual. Saf. Eng. 6, 103–122 (1999)
39. 3.39 W. Nelson: Accelerated Testing (Wiley, New York 1990)
40. 3.40 T. Nakagawa, S. Osaki: The discrete Weibull distribution, IEEE Trans. Reliab. 24, 300–301 (1975)
41. 3.41 W. E. Stein, R. Dattero: A new discrete Weibull distribution, IEEE Trans. Reliab. 33, 196–197 (1984)
42. 3.42 W. J. Padgett, J. D. Spurrier: Discrete failure models, IEEE Trans. Reliab. 34, 253–256 (1985)
43. 3.43 W. M. Bassin: A Bayesian optimal overhaul model for the Weibull restoration process, J. Am. Stat. Ass. 68, 575–578 (1973)
44. 3.44 L. J. Bain, M. Engelhart: Statistical Analysis of Reliability and Life-testing Models, 2nd edn. (Mar-cel Dekker, New York 1991)
45. 3.45 D. R. Cox, D. D. Oakes: Analysis of Survival Data (Chapman Hall, New York 1984)
46. 3.46 N. Yannaros: Weibull renewal processes, Ann. Inst. Stat. Math. 46, 641–648 (1994)
47. 3.47 D. N. P. Murthy, M. Bulmer, J. E. Eccleston: Weibull modelling, Reliab. Eng. Sys. Saf. 86, 257–267 (2004)
48. 3.48 W. Q. Meeker, L. A. Escobar: Statistical Methods for Reliability Data (Wiley, New York 1998)
49. 3.49 K. Keshevan, G. Sargent, H. Conrad: Statistical analysis of the Hertzian fracture of pyrex glass using the Weibull distribution function, J. Mater. Sci. 15, 839–844 (1980)
50. 3.50 A. K. Sheikh, J. K. Boah, D. A. Hansen: Statistical modelling of pitting corrosion and pipeline reliability, Corrosion 46, 190–196 (1990)
51. 3.51 F. S. Queeshi, A. K. Sheikh: Probabilistic characterization of adhesive wear in metals, IEEE Trans. Reliab. 46, 38–44 (1997)
52. 3.52 S. D. Durham, W. J. Padgett: Cumulative damage model for system failure with application to carbon fibers and composites, Technometrics 39, 34–44 (1997)
53. 3.53 J. B. Almeida: Application of Weilbull statistics to the failure of coatings, J. Mater. Proces. Technol. 93, 257–263 (1999)
54. 3.54 S. L. Fok, B. C. Mitchell, J. Smart, B. J. Marsden: A numerical study on the application of the Weibull theory to brittle materials, Eng. Fract. Mech. 68, 1171–1179 (2001)
55. 3.55 J. A. Newell, T. Kurzeja, M. Spence, M. Lynch: Analysis of recoil compressive failure in high per-formance polymers using two-, four-parameter Weibull models, High Perform. Polym. 14, 425–434 (2002)
56. 3.56 Q. S. Li, J. Q. Fang, D. K. Liu, J. Tang: Failure probability prediction of concrete components, Cement Concrete Res. 33, 1631–1636 (2003)
57. 3.57 J. T. Duane: Learning curve approach to reliability monitoring, IEEE Trans. Aerosp. 40, 563–566 (1964)
58. 3.58 L. H. Crow: Reliability analysis for complex systems. In: Reliability and Biometry, ed. by F. Proschan, R. J. Serfling (SIAM, Philadelphia 1974) pp. 379– 410
59. 3.59 M. A. Rahim: Economic design of X charts assuming Weibull in-control times, J. Qual. Technol. 25, 296– 305 (1993)
60. 3.60 A. F. B. Costa, M. A. Rahim: Economic design of X and R charts under Weibull shock models, Qual. Reliab. Eng. Int. 16, 143–156 (2000)
61. 3.61 Y. S. Chen, Y. M. Yang: Economic design of x-control charts with Weibull in-control times, when there are multiple assignable causes, Int. J. Prod. Econ. 77, 17–23 (2002)
62. 3.62 P. R. Nelson: Control chart for Weibull process, IEEE Trans. Reliab. 28, 283–288 (1979)
63. 3.63 D. N. P. Murthy, I. Djamaludin, R. J. Wilson: Product warranty and quality control, Qual. Reliab. Eng. 9, 431–443 (1993)
64. 3.64 I. Djamaludin, D. N. P. Murthy, R. J. Wilson: Lot-sizing and testing for items with uncertain quality, Math. Comp. Model. 22, 35–44 (1995)
65. 3.65 D. Kececioglu, F. B. Sun: Burn-in Testing: Its Quantification and Optimization (Prentice Hall, Upper Saddle River 1997)
66. 3.66 W. R. Blischke, D. N. P. Murthy: Product Warranty Handbook (Marcel Dekker, New York 1994)
67. 3.67 W. R. Blischke, D. N. P. Murthy: Warranty Cost Analysis (Marcel Dekker, New York 1996)
68. 3.68 D. N. P. Murthy, I. Djamaludin: Product warranty: A review, Int. J. Prod. Econ. 79, 231–260 (2002)
69. 3.69 P. R. Tadikmalla: Age replacement policies for Weibull failure rates, IEEE Trans. Reliab. 29, 88–90 (1980)
70. 3.70 W. R. Blischke, D. N. P. Murthy: Reliability (Wiley, New York 2000)
71. 3.71 M. Al-Hasan, R. R. Nigmatullin: Identification of the generalized Weibull distribution in wind speed data by the Eigen-coordinates method, Renewable En-erg. 28(1), 93–110 (2003)
72. 3.72 T. Huillet, H. F. Raynaud: Rare events in a log-Weibull scenario—Application to earthquake magnitude data, Eur. Phys. J. B 12, 457–469 (1999)
73. 3.73 M. S. Bebbington, C. D. Lai: On nonhomogeneous models for volcanic eruptions, Math. Geol. 28, 585– 599 (1996)
74. 3.74 S. R. Durrans: Low-flow analysis with aconditional Weibull tail model, Water Resour. Res. 32, 1749–1760 (1996)
75. 3.75 J. H. Heo, D. C. Boes, J. D. Salas: Regional flood frequency analysis based on a Weibull model: Part 1. Estimation, asymptotic variances, J. Hydrol. 242, 157–170 (2001)
76. 3.76 P. Mafart, O. Couvert, S. Gaillard, I. Leguerinel: On calculating sterility in thermal preservation methods: application of the Weibull frequency distribution model, Int. J. Food Microbiol. 72, 107–113 (2002)
77. 3.77 K. Roed, T. Zhang: A note on the Weibull distribution and time aggregation bias, Appl. Econ. Lett. 9, 469– 472 (2002)
78. 3.78 H. Dahm, J. Niemeyer, D. Schroder: Application of the Weibull distribution to describe the vertical distribution of cesium-137 on a slope under permanent pasture in Luxembourg, J. Environ. Radioac. 63, 207–219 (2002)
79. 3.79 R. A. Fleming: The Weibull model and an ecological application: describing the dynamics of foliage biomass on Scots pine, Ecol. Model. 138(1-3), 309– 319 (2001)
80. 3.80 K. J. Carroll: On the use and utility of the Weibull model in the analysis of survival data, Control. Clin. Trials 24, 682–701 (2003)
81. 3.81 N. L. Johnson, S. Kotz, N. Balakrishnan: Continuos Univariate Distributions, Vol. 1, 2nd edn. (Wiley, New York 1994) pp. 657–661