Bias and standard deviation due to Weibull parameter estimation for small data sets

Weibull statistics have become the most popular statistics to describe breakdown events in solid dielectrics. The version most applied is probably the Weibull 2-parameter model. The present paper briefly reviews the development of the Weibull distribution while paying most attention to the 2-parameter version. Subsequently, the performance of two types of analysis methods, i.e. maximum likelihood and linear regression methods, are investigated. Particularly, the performance with small data sets were of interest. The reason for this interest are common events that do not allow the collection of larger data sets. Rather than fall into pure gambling, it is attempted to describe both the bias, i.e. systematic error, and the standard deviation under these conditions. As a practical tool, a recently introduced asymptotic difference function is utilized to cast bias and standard deviation in simple formulas, even for small data sets. The unbiased maximum likelihood method is also compared to an alternative so-called 'generalized' or 'relative' maximum likelihood method. For ten cases, a selection of formulas is tabulated which can be added to current programs. Among this set is a formula to unbias the Weibull shape parameter for censored data sets.