Correction Factor for Unbiased Estimation of Weibull Modulus by the Linear Least Squares Method

  • Xiang JiaEmail author
  • Guoguo Xi
  • Saralees Nadarajah


In material science, linear least squares is the most popular method to estimate Weibull parameters for stress data. However, the estimation \( {\hat{m}} \) of the Weibull modulus m is usually biased due to the data uncertainty and shorcoming of estimation methods. Many researchers have developed techniques to produce unbiased estimation of m. In this study, a correction factor is considered. First, the distribution of \( {\hat{m}} \) is derived mathematically and proved through a Monte Carlo simulation numerically again. Second, based on the derived distribution, the correction factor that depends only on the probability estimator of cumulative failure and stress data size is presented. Then, simple procedures are proposed to compute it. Further, the correction factors for four common probability estimators and typical sizes are displayed. The coefficient of variation and mode are also discussed to determine the optimal probability estimator. Finally, the proposed correction factor is applied to four groups of stress data for the unbiased estimation of m correspondingly concerning the alumina agglomerate, ball stud, coated conductor and steel, respectively.



The authors would like to thank the editor and referees for careful reading and comments which greatly improved the paper. This work was partially supported by the National Natural Science Foundation of China under Grant no. 71801219 and the Hunan Provincial Natural Science Foundation of China 2019JJ50730.


  1. 1.
    I. Davies, T. Ishikawa, M. Shibuya and T. Hirokawa: Compos. Sci. Technol., 1999, vol. 59, pp. 801–811.CrossRefGoogle Scholar
  2. 2.
    I. Davies, T. Ishikawa, M. Shibuya, T. Hirokawa and J. Gotoh: Compos Part A., 1999, vol. 30, pp. 587–591.CrossRefGoogle Scholar
  3. 3.
    A. Badu and V, Jayabalan. J. Mater. Sci. Technol., vol. 25, pp. 341-343 (2009)Google Scholar
  4. 4.
    Y. Boiko. Colloid. Polym. Sci., 2017, vol. 295, pp. 1993–1999.CrossRefGoogle Scholar
  5. 5.
    J. Quinn and G. Quinn. Dent. Mater., 2010, vol. 26, pp. 135–147.CrossRefGoogle Scholar
  6. 6.
    D. Wu and J. Zhou and Y. Li: J. Eur. Ceram. Soc., 2006, vol. 26, pp. 1099–1105.CrossRefGoogle Scholar
  7. 7.
    A. Khalili and K. Krom: J. Mater. Sci., 1991, vol. 26, pp. 6741–6752.CrossRefGoogle Scholar
  8. 8.
    D. Wu, Y.Li, J. Zhang, L.Chang, D.Wu, Z.Fang, and Y. Shi: Chem. Eng. Sci., 2001, vol. 56, pp. 7035–7044.CrossRefGoogle Scholar
  9. 9.
    K. Trustrum, A. De, S. Jayatilaka, J. Mater. Sci. 14, 1080–1084 (1979)CrossRefGoogle Scholar
  10. 10.
    D. Wu, G.Lu, H.Jiang and Y. Li: J. Am. Ceram. Soc., 2006, vol. 26, pp. 1099–1105.CrossRefGoogle Scholar
  11. 11.
    N. Hua, G. Li, C. Lin, X. Ye, W. Wang and W. Chen: J. Non-cryst. Solids., 2015, vol. 432, pp. 342–347.CrossRefGoogle Scholar
  12. 12.
    A. Talimian and V M. Sglavo: J. Non-cryst. Solids., 2017, vol. 456, pp. 12–21.CrossRefGoogle Scholar
  13. 13.
    A S. Haidyrah, J W. Newkirk, and C H. Castao: J. Nucl. Mater., 2016, vol. 470, pp. 244–250.CrossRefGoogle Scholar
  14. 14.
    G. Ma, W. Zhou, R A. Regueiro, Q. Wang and X. Chang: Powder. Technol., 2017, vol. 308, pp. 388–397.CrossRefGoogle Scholar
  15. 15.
    L. Yang, P. Cai, Z. Xu, Y. Jin, C. Liang, F. Yin and T. Zhai: Int. J. Fatigue., 2016, vol. 96, pp. 185–195.CrossRefGoogle Scholar
  16. 16.
    G. Quercia, D. Chan and K. Luke: J. Petrol. Sci. Eng., 2016, vol. 146, pp. 536–544.CrossRefGoogle Scholar
  17. 17.
    Z. Lv, P. Cai, T. Yu, Y. Jin, H. Zhang, W. Fu and T. Zhai: J. Alloy. Compd., 2017, vol. 691, pp. 103–109.CrossRefGoogle Scholar
  18. 18.
    B. Chen, X. Zhang, J. Yu and Y. Wang: Constr. Bulid. Materj., 2017, vol. 133, pp. 330–339.CrossRefGoogle Scholar
  19. 19.
    I. Davies: J. Mater. Sci., 2004, vol. 39, pp. 1441–1444.CrossRefGoogle Scholar
  20. 20.
    L. Zhang, M. Xie and L. Tang: Qual. Reliab. Eng. Int., 2006, vol. 22, pp. 905–917.CrossRefGoogle Scholar
  21. 21.
    I. Davies: J. Eur. Ceram. Soc., 2017, vol. 37, pp. 369–380.CrossRefGoogle Scholar
  22. 22.
    I. Davies: J. Eur. Ceram. Soc., 2017, vol. 37, pp. 2973–2981.CrossRefGoogle Scholar
  23. 23.
    I. Davies: J. Mater. Sci. Lett., 2001, vol. 20, pp. 997–999.CrossRefGoogle Scholar
  24. 24.
    X. Jia, P. Jiang and B. Guo: J. Cent. South Univ., 2015, vol. 22, pp. 3506–3511.CrossRefGoogle Scholar
  25. 25.
    D. Thoman, L. Bain and C. Antle: Technometrics, 1969, vol. 11, pp. 445–460.CrossRefGoogle Scholar
  26. 26.
    M. Matsumoto and T. Nishimura: ACM T. Model. Comput. S., 1998, vol. 8, pp. 3–30.CrossRefGoogle Scholar
  27. 27.
    X. Jia, S. Nadarajah and B. Guo: IEEE T. Reliab., 2018, vol. 67, pp. 432–445.CrossRefGoogle Scholar
  28. 28.
    I. Park, K. Nam and C. Kang: J. Mech. Sci. Technol., 2018, vol. 32, pp. 5647–5652.CrossRefGoogle Scholar
  29. 29.
    S. Muto, S. Fujita, K. Akashi and T. Yoshida et al.: IEEE Trans. Appl. Supercon., 2018, vol. 28, pp. 1–4.CrossRefGoogle Scholar
  30. 30.
    A. Tiwari, A. Gopalan, A. Shokry, R. Singh and P. Stahle: Int. J. Fract., 2017, vol. 205, pp. 103–109.CrossRefGoogle Scholar

Copyright information

© The Minerals, Metals & Materials Society and ASM International 2019

Authors and Affiliations

  1. 1.College of Systems EngineeringNational University of Defense TechnologyHunanP.R. China
  2. 2.School of Materials Science and EngineeringBeihang UniversityBeijingP.R. China
  3. 3.School of MathematicsUniversity of ManchesterManchesterUK

Personalised recommendations