Template:Characteristics of the generalized gamma distribution: Difference between revisions

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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.
 
[[Image:WB chp12 pdf.png|center|250px| ]]  
 
:• The Weibull distribution is a special case when  <math>\lambda =1</math>  and:
 
::<math>\begin{align}
  & \beta = \frac{1}{\sigma } \\
& \eta = \ln (\mu ) 
\end{align}</math>
 
:• 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
<1,</math>  <math>\beta =1,</math>  and  <math>\beta >1</math>  respectively).
:• The exponential distribution is a special case when  <math>\lambda =1</math>  and  <math>\sigma =1</math>.
:• The lognormal distribution is a special case when  <math>\lambda =0</math>.
:• The gamma distribution is a special case when  <math>\lambda =\sigma </math>.
 
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>.

Latest revision as of 09:35, 9 August 2012