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

In material science, linear least squares is the most popular method to estimate Weibull parameters for stress data. However, the estimation 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 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.