Improving the 1-parameter Weibull: A Bayesian approach

Using maximum likelihood estimation (MLE) to estimate the parameters in a Weibull distribution will lead to a biased estimation of the shape parameter when the sample size is small or too few failures are observed. This bias may lead to inaccurate reliability point estimates. In addition, with few data points available in the calculation, the uncertainty of the estimated parameters is high, which again leads to high uncertainty in the predicted reliability (i.e. wide confidence bounds). To overcome these two issues, the 1-parameter Weibull distribution has been widely used, provided that the shape parameter is known beforehand. This approach, however, does not account for any uncertainty in the assumed value of the shape parameter and can therefore yield optimistic results in the form of tight confidence bounds. It can be improved with better information about the variability of the shaper parameter. In this paper, a Bayesian model, which is an improved approach for the 1-parameter Weibull, is discussed. Recommendations for establishing variability models for the Weibull shape parameter are presented.