Advertisement

Journal of Materials Science

, Volume 42, Issue 24, pp 10173–10179 | Cite as

On estimating Weibull modulus by the linear regression method

  • Murat TiryakioğluEmail author
  • David Hudak
Article

Abstract

Statistical models were developed to estimate the bias of the shape parameter of a 2-parameter Weibull distribution where the shape parameter was estimated using a linear regression. These models were formulated for 27 sample sizes from 5 to 100 and for 35 probability estimators, \( P = {(i - a)} \mathord{\left/ {\vphantom {{i - a} {n + b}}} \right. \kern-\nulldelimiterspace} {(n + b)} \), by varying “a” and “b”. In each simulation, 20,000 trials were used. From these models, a class of unbiased estimators were developed for each sample size. The standard deviation and coefficient of variation of these estimators were compared to the bias of the estimators. The standard deviation increased while the coefficient of variation decreased with increasing bias of the shape parameter. Also, the Anderson–Darling statistics was used to determine that the normal, log-normal, 3-parameter Weibull, and 3-parameter log-Weibull distributions did not provide good fit to the estimator of the shape parameter.

Keywords

Shape Parameter Brittle Fracture Lognormal Distribution Weibull Distribution Maximum Likelihood Method 

References

  1. 1.
    Khalili A, Kromp K (1991) J Mater Sci 26:6741CrossRefGoogle Scholar
  2. 2.
    Langlois R (1991) J Mater Sci Lett 10:1049CrossRefGoogle Scholar
  3. 3.
    Trustrum K, Jayatilaka AdeS (1979) J Mater Sci 14:1080CrossRefGoogle Scholar
  4. 4.
    Hazen A (1914) Trans ASCE 77:1547Google Scholar
  5. 5.
    Weibull W (1939) Ingeniörsvetenskapsakademiens Handlingar Nr 151Google Scholar
  6. 6.
    Benard A, Bosi-Levenbach ED (1953) Statistica 7:163CrossRefGoogle Scholar
  7. 7.
    Blom G (1958) Statistical estimates of transformed beta variables. Wiley, NY, pp 68–75, 143–146Google Scholar
  8. 8.
    Gringorten II (1963) J Geophysical Res 68:813CrossRefGoogle Scholar
  9. 9.
    Adamowski K (1981) Water Resources Bull 17:197CrossRefGoogle Scholar
  10. 10.
    Cunane C (1978) J Hydrology 37:205CrossRefGoogle Scholar
  11. 11.
    Tukey JW (1962) Annals of Math Stat 33:1CrossRefGoogle Scholar
  12. 12.
    Beard LR (1943) Trans ASCE 69:1110Google Scholar
  13. 13.
    Wu D, Zhoua J, Li Y (2006) J Eur Cer Soc 26:1099CrossRefGoogle Scholar
  14. 14.
    Tiryakioğlu M (2006) J Mater Sci 41:5011CrossRefGoogle Scholar
  15. 15.
    Davies IJ (2001) J Mater Sci Lett 20:997CrossRefGoogle Scholar
  16. 16.
    Gong J (2000) J Mater Sci Lett 19:827CrossRefGoogle Scholar
  17. 17.
    Ritter J, Bandyopadhyay N, Jakus K (1981) Am Cer Soc Bull 60:788Google Scholar
  18. 18.
    Bergman B (1984) J Mater Sci Lett 3:689CrossRefGoogle Scholar
  19. 19.
    Gong J, Wang J (2002) Key Eng Mater 224–226:779CrossRefGoogle Scholar
  20. 20.
    Barbero E, Fernandez-Saez J, Navarro C (2000) Composites: Part B 31:375CrossRefGoogle Scholar
  21. 21.
    Barbero E, Fernandez-Saez J, Navarro C (2001) J Mater Sci Lett 20:847CrossRefGoogle Scholar
  22. 22.
    Tiryakioğlu M (2007) J Mater Sci, doi:10.1007/s10853-007-2095-7CrossRefGoogle Scholar
  23. 23.
    Anderson TW, Darling DA (1954) J Am Stat Assoc 49:765CrossRefGoogle Scholar
  24. 24.
    Stephens MA (1974) J Am Stat Assoc 69:730CrossRefGoogle Scholar
  25. 25.
    Stephens MA (1986) In: D’Agostino RB, Stephens MA (eds) Goodness of fit techniques. Marcel Dekker, p 97Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Engineering, School of Engineering, Mathematics and Science Robert Morris UniversityMoon TownshipUSA
  2. 2.Department of MathematicsSchool of Engineering, Mathematics and Science, Robert Morris UniversityMoon TownshipUSA

Personalised recommendations