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| ===Characteristics of the Generalized Gamma Distribution===
| | #REDIRECT [[The_Generalized_Gamma_Distribution]] |
| As mentioned previously, the generalized gamma distribution includes other distributions as special cases based on the values of the parameters.
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| [[Image:ldagamma10.1.gif|thumb|center|500px| ]] | |
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| :• The Weibull distribution is a special case when <math>\lambda =1</math> and:
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| ::<math>\begin{align}
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| & \beta = & \frac{1}{\sigma } \\
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| & \eta = & \ln (\mu )
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| \end{align}</math>
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| :• In this case, the generalized distribution has the same behavior as the Weibull for <math>\sigma >1,</math> <math>\sigma =1,</math> and <math>\sigma <1</math> ( <math>\beta
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| <1,</math> <math>\beta =1,</math> and <math>\beta >1</math> respectively).
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| :• The exponential distribution is a special case when <math>\lambda =1</math> and <math>\sigma =1</math>.
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| :• The lognormal distribution is a special case when <math>\lambda =0</math>.
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| :• The gamma distribution is a special case when <math>\lambda =\sigma </math>.
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| By allowing <math>\lambda </math> to take negative values, the generalized gamma distribution can be further extended to include additional distributions as special cases. For example, the Fréchet distribution of maxima (also known as a reciprocal Weibull) is a special case when <math>\lambda =-1</math>.
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