Research Article
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Year 2025, Early View, 1 - 1
https://doi.org/10.35378/gujs.1306771

Abstract

References

  • [1] Weibull, W., “A statistical distribution function of wide applicability”, Journal of Applied Mechanics, (1951).
  • [2] Lai, C.D., Murthy D., Xie, M., “Weibull distributions and their applications”, Springer Handbooks, Springer, 63–78, (2006).
  • [3] Rinne, H., “The Weibull distribution: a handbook”, Chapman and Hall/CRC, (2008).
  • [4] Barbero, E., Fernández-Sáez, J., Navarro, C., “Statistical analysis of the mechanical properties of composite materials”, Composites Part B: Engineering, 31: 375–81, (2000).
  • [5] Yang, X., Xie, L., Yang, Y., Zhao, B., Li, Y., “A comparative study for parameter estimation of the Weibull distribution in a small sample size: An application to spring fatigue failure data”, Quality Engineering, 1-13, (2022).
  • [6] Babacan, E.K., Kaya, S., “A simulation study of the Bayes estimator for parameters in Weibull distribution”, Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics, 68:1664–74, (2019).
  • [7] Babacan, E.K., Kaya, S., “Comparison of parameter estimation methods in Weibull Distribution”, Sigma Journal of Engineering and Natural Sciences, 38: 1609–21, (2020).
  • [8] Genschel, U., Meeker, W.Q., “A comparison of maximum likelihood and median-rank regression for Weibull estimation”, Quality Engineering, 22: 236–55, (2010).
  • [9] Jia, X., Xi, G., Nadarajah, S., “Correction Factor for Unbiased Estimation of Weibull Modulus by the Linear Least Squares Method”, Metallurgical and Materials Transactions A, 50: 2991–3001, (2019).
  • [10] Davies, I.J., “Unbiased estimation of Weibull modulus using linear least squares analysis—A systematic approach”, Journal of the European Ceramic Society, 37: 369–80, (2017).
  • [11] Kantar, Y.M., “Generalized least squares and weighted least squares estimation methods for distributional parameters”, REVSTAT-Statistical Journal, 13: 263–82, (2015).
  • [12] Wu, D., Lu, G., Jiang, H., Li, Y., “Improved estimation of Weibull parameters with the linear regression method”, Journal of the American Ceramic Society, 87: 1799–802, (2004).
  • [13] Wu, D., Zhou, J., Li, Y., “Unbiased estimation of Weibull parameters with the linear regression method”, Journal of the European Ceramic Society, 26: 1099–105, (2006).
  • [14] Tiryakioğlu, M., “An unbiased probability estimator to determine Weibull modulus by the linear regression method”, Journal of Materials Science, 41: 5011–3, (2006).
  • [15] Tiryakioğlu, M., Hudak, D., “On estimating Weibull modulus by the linear regression method”, Journal of Materials Science, 42: 10173–9, (2007).
  • [16] Davies, I.J., “Unbiased estimation of the Weibull scale parameter using linear least squares analysis”, Journal of the European Ceramic Society, 37: 2973–81, (2017).
  • [17] Song, L., Wu, D., Li, Y., “Optimal probability estimators for determining Weibull parameters”, Journal of Materials Science Letters, 22: 1651–1653, (2003).
  • [18] D’Agostino, R., “Goodness-of-fit-techniques”, Routledge, (2017).
  • [19] Dutang C, Wuertz, D., “A note on random number generation”. https://cran.r-project.org/web/packages/randtoolbox/vignettes/fullpres.pdf. Access date: 30.06.2024.
  • [20] Bütikofer, L., Stawarczyk, B., Roos, M., “Two regression methods for estimation of a two-parameter Weibull distribution for reliability of dental materials”, Dental Materials, 31: 33–50, (2015).
  • [21] Fernandez-Saez, J., Chao, J., Duran, J., Amo, J.M., “Estimating lower-bound fracture parameters for brittle materials”, Journal of Materials Science Letters, 12: 1493–6, (1993).

A New Rank Estimator for Least Squares Estimation of Weibull Modulus

Year 2025, Early View, 1 - 1
https://doi.org/10.35378/gujs.1306771

Abstract

The Weibull distribution is widely used in reliability analysis to evaluate the failure behavior and lifetime characteristics of various systems and components. One of the most commonly used methods for estimating the parameters of the Weibull distribution is the ordinary least squares (OLS) technique, which is based on fitting a linear regression model to the transformed data. This paper proposes a new rank estimator for ordinary least squares estimation of Weibull modulus, a key parameter used as a measure of variability in the data. The new rank estimator is a quadratic function of the ranks of order statistics, with three parameters that are optimized by Monte Carlo simulations. Using relative efficiency as a criterion, the performance of the new rank estimator is compared with three commonly used rank estimators, mean, median and Hazen rank estimators, which are linear functions of the ranks of order statistics. The results show that the new rank estimator has a significant advantage over the other rank estimators for any sample size between 3 and 150. The findings also imply that other nonlinear functions, such as cubic polynomials, could be applied to further improve the efficiency of the parameter estimators of the ordinary least squares method.

Supporting Institution

This research is not supported by any institution

References

  • [1] Weibull, W., “A statistical distribution function of wide applicability”, Journal of Applied Mechanics, (1951).
  • [2] Lai, C.D., Murthy D., Xie, M., “Weibull distributions and their applications”, Springer Handbooks, Springer, 63–78, (2006).
  • [3] Rinne, H., “The Weibull distribution: a handbook”, Chapman and Hall/CRC, (2008).
  • [4] Barbero, E., Fernández-Sáez, J., Navarro, C., “Statistical analysis of the mechanical properties of composite materials”, Composites Part B: Engineering, 31: 375–81, (2000).
  • [5] Yang, X., Xie, L., Yang, Y., Zhao, B., Li, Y., “A comparative study for parameter estimation of the Weibull distribution in a small sample size: An application to spring fatigue failure data”, Quality Engineering, 1-13, (2022).
  • [6] Babacan, E.K., Kaya, S., “A simulation study of the Bayes estimator for parameters in Weibull distribution”, Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics, 68:1664–74, (2019).
  • [7] Babacan, E.K., Kaya, S., “Comparison of parameter estimation methods in Weibull Distribution”, Sigma Journal of Engineering and Natural Sciences, 38: 1609–21, (2020).
  • [8] Genschel, U., Meeker, W.Q., “A comparison of maximum likelihood and median-rank regression for Weibull estimation”, Quality Engineering, 22: 236–55, (2010).
  • [9] Jia, X., Xi, G., Nadarajah, S., “Correction Factor for Unbiased Estimation of Weibull Modulus by the Linear Least Squares Method”, Metallurgical and Materials Transactions A, 50: 2991–3001, (2019).
  • [10] Davies, I.J., “Unbiased estimation of Weibull modulus using linear least squares analysis—A systematic approach”, Journal of the European Ceramic Society, 37: 369–80, (2017).
  • [11] Kantar, Y.M., “Generalized least squares and weighted least squares estimation methods for distributional parameters”, REVSTAT-Statistical Journal, 13: 263–82, (2015).
  • [12] Wu, D., Lu, G., Jiang, H., Li, Y., “Improved estimation of Weibull parameters with the linear regression method”, Journal of the American Ceramic Society, 87: 1799–802, (2004).
  • [13] Wu, D., Zhou, J., Li, Y., “Unbiased estimation of Weibull parameters with the linear regression method”, Journal of the European Ceramic Society, 26: 1099–105, (2006).
  • [14] Tiryakioğlu, M., “An unbiased probability estimator to determine Weibull modulus by the linear regression method”, Journal of Materials Science, 41: 5011–3, (2006).
  • [15] Tiryakioğlu, M., Hudak, D., “On estimating Weibull modulus by the linear regression method”, Journal of Materials Science, 42: 10173–9, (2007).
  • [16] Davies, I.J., “Unbiased estimation of the Weibull scale parameter using linear least squares analysis”, Journal of the European Ceramic Society, 37: 2973–81, (2017).
  • [17] Song, L., Wu, D., Li, Y., “Optimal probability estimators for determining Weibull parameters”, Journal of Materials Science Letters, 22: 1651–1653, (2003).
  • [18] D’Agostino, R., “Goodness-of-fit-techniques”, Routledge, (2017).
  • [19] Dutang C, Wuertz, D., “A note on random number generation”. https://cran.r-project.org/web/packages/randtoolbox/vignettes/fullpres.pdf. Access date: 30.06.2024.
  • [20] Bütikofer, L., Stawarczyk, B., Roos, M., “Two regression methods for estimation of a two-parameter Weibull distribution for reliability of dental materials”, Dental Materials, 31: 33–50, (2015).
  • [21] Fernandez-Saez, J., Chao, J., Duran, J., Amo, J.M., “Estimating lower-bound fracture parameters for brittle materials”, Journal of Materials Science Letters, 12: 1493–6, (1993).
There are 21 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Article
Authors

Burak Birgören 0000-0001-9045-6092

Early Pub Date September 26, 2024
Publication Date
Published in Issue Year 2025 Early View

Cite

APA Birgören, B. (2024). A New Rank Estimator for Least Squares Estimation of Weibull Modulus. Gazi University Journal of Science1-1. https://doi.org/10.35378/gujs.1306771
AMA Birgören B. A New Rank Estimator for Least Squares Estimation of Weibull Modulus. Gazi University Journal of Science. Published online September 1, 2024:1-1. doi:10.35378/gujs.1306771
Chicago Birgören, Burak. “A New Rank Estimator for Least Squares Estimation of Weibull Modulus”. Gazi University Journal of Science, September (September 2024), 1-1. https://doi.org/10.35378/gujs.1306771.
EndNote Birgören B (September 1, 2024) A New Rank Estimator for Least Squares Estimation of Weibull Modulus. Gazi University Journal of Science 1–1.
IEEE B. Birgören, “A New Rank Estimator for Least Squares Estimation of Weibull Modulus”, Gazi University Journal of Science, pp. 1–1, September 2024, doi: 10.35378/gujs.1306771.
ISNAD Birgören, Burak. “A New Rank Estimator for Least Squares Estimation of Weibull Modulus”. Gazi University Journal of Science. September 2024. 1-1. https://doi.org/10.35378/gujs.1306771.
JAMA Birgören B. A New Rank Estimator for Least Squares Estimation of Weibull Modulus. Gazi University Journal of Science. 2024;:1–1.
MLA Birgören, Burak. “A New Rank Estimator for Least Squares Estimation of Weibull Modulus”. Gazi University Journal of Science, 2024, pp. 1-1, doi:10.35378/gujs.1306771.
Vancouver Birgören B. A New Rank Estimator for Least Squares Estimation of Weibull Modulus. Gazi University Journal of Science. 2024:1-.