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

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Abstract

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.

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Acknowledgments

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.

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Authors and Affiliations

  1. College of Systems Engineering, National University of Defense Technology, Hunan, 410073, P.R. China

    Xiang Jia

  2. School of Materials Science and Engineering, Beihang University, Beijing, 100191, P.R. China

    Guoguo Xi

  3. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Saralees Nadarajah

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  1. Xiang Jia
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  2. Guoguo Xi
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  3. Saralees Nadarajah
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Correspondence to Xiang Jia.

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Manuscript submitted October 20, 2018.

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Jia, X., Xi, G. & Nadarajah, S. Correction Factor for Unbiased Estimation of Weibull Modulus by the Linear Least Squares Method. Metall Mater Trans A 50, 2991–3001 (2019). https://doi.org/10.1007/s11661-019-05216-x

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  • DOI: https://doi.org/10.1007/s11661-019-05216-x

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