Unbiased estimation of Weibull modulus using linear least squares analysis-A systematic approach

The wide applicability of the Weibull distribution to fields such as hydrology and materials science has led to a large number of probability estimators being proposed, in particular for the widely used technique of obtaining the Weibull modulus, m, using unweighted linear least squares (LLS) analysis. In this work a systematic approach using the Monte Carlo method has been taken to determining the optimal probability estimators for unbiased estimation of m (mean, median and mode) using the general equation F=(i−a)/(N+b) whilst simultaneously minimising the coefficient of variation for each of the average values. A wide range of a and b values were investigated within the region 0≤a≤1 and 1≤b≤1000 with the form of F=(i−a)/(N+1) being chosen as the recommend probability estimator equation due to its simplicity and relatively small coefficient of variation. Values of a as a function of N were presented for the mean, median and mode m values.