Shapes, Moments and Estimators of the Weibull Distribution

This presentation describes, in more detail than heretofore published, the properties of the Weibull distribution using the following equation: f (t; ¿, ß, ¿) = ß/¿ß(t - ¿)ß-1 exp[-(t-¿/¿)ß],t >¿, where ¿ > 0 is a scale parameter in time units ß > 0 is a shape parameter (dimensionless) ¿ (any real value) is a location parameter in time units. Conditions on the shape parameter ß for the existence of a mode and inflection point are given; the locations of and the values of the function at these points are traced as ß grows from zero to infinity. The behavior of the median and first four moments is described and presented in tabular form as a function of ß. Other interesting features of this distribution are noted. Finally, given the fatilure times of n randomly selected units, the maximum likelihood equations are derived. When solved by machine computer methods, these equations will give approximations for the maximum likelihood estimators of the three parameters.