Uncertainty Evaluation of Weibull Estimators through Monte Carlo Simulation: Applications for Crack Initiation Testing

Abstract

The typical experimental procedure for testing stress corrosion cracking initiation involves an interval-censored reliability test. Based on these test results, the parameters of a Weibull distribution, which is a widely accepted crack initiation model, can be estimated using maximum likelihood estimation or median rank regression. However, it is difficult to determine the appropriate number of test specimens and censoring intervals required to obtain sufficiently accurate Weibull estimators. In this study, we compare maximum likelihood estimation and median rank regression using a Monte Carlo simulation to examine the effects of the total number of specimens, test duration, censoring interval, and shape parameters of the true Weibull distribution on the estimator uncertainty. Finally, we provide the quantitative uncertainties of both Weibull estimators, compare them with the true Weibull parameters, and suggest proper experimental conditions for developing a probabilistic crack initiation model through crack initiation tests.

Effects of Cracking Test Conditions on Estimation Uncertainty for Weibull Parameters Considering Time-Dependent Censoring Interval

“…Additionally, critical lines were observed after which exceedingly wide ${\mathrm{RLCI}}_{90\%}(\stackrel{false^}{\mathrm{sans-serif\beta }})$ were produced [12]. The gradients of ${\mathrm{RLCI}}_{90\%}(\stackrel{false^}{\mathrm{sans-serif\beta }})$ were very high near the critical lines.…”

“…Thus, the test duration was considered as a factor of estimation uncertainties in an earlier study [12]. However, the results indicated that there were deficiencies to using the test duration as a factor of estimation uncertainties.…”

“…First, experimenters did not know their relative test duration (RTD), which is defined as follows: $\mathrm{RTD}=\frac{testduration}{{\mathsf{\eta }}_{\mathrm{true}}},$ this is because the experimenters did not know the exact value of ${\mathsf{\eta }}_{\mathrm{true}}$. Additionally, it was possible to continue the simulation even after all of the specimens cracked when the test duration was used as a fixed input for a simulation study [12]. This is an undesirable phenomenon for the MRR estimation because it produces redundant and often irrelevant results close to the end time of the simulation.…”

“…Therefore, in our previous study [12], the effects of certain experimental conditions on estimation uncertainties of Weibull parameters were investigated through the Monte Carlo simulation. However, the study also revealed that the following issues warranted further investigation:

In the previous study, it was assumed that LCI was a time-independent variable.

…”

“…Additionally, it was possible to continue the simulation even after all of the specimens cracked if the test duration was used as a fixed input for the simulation study [12]. …”