Shapes, Moments and Estimators of the Weibull Distribution

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Abstract
This presentation describes, in more detail than heretofore published, the properties of the Weibull distribution using the following equation: f (t; α, ß, γ) = ß/αß(t - γ)ß-1 exp[-(t-γ/α)ß],t >γ, where α > 0 is a scale parameter in time units ß > 0 is a shape parameter (dimensionless) γ (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.

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