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

From ReliaWiki
Jump to navigation Jump to search
(Created page with '===Characteristics of the Generalized Gamma Distribution=== As mentioned previously, the generalized gamma distribution includes other distributions as special cases based on the…')
 
Line 7: Line 7:


::<math>\begin{align}
::<math>\begin{align}
   & \beta = & \frac{1}{\sigma } \\  
   & \beta = \frac{1}{\sigma } \\  
  & \eta = & \ln (\mu )   
  & \eta = & \ln (\mu )   
\end{align}</math>
\end{align}</math>

Revision as of 21:47, 14 February 2012

Characteristics of the Generalized Gamma Distribution

As mentioned previously, the generalized gamma distribution includes other distributions as special cases based on the values of the parameters.

Ldagamma10.1.gif
• The Weibull distribution is a special case when [math]\displaystyle{ \lambda =1 }[/math] and:
[math]\displaystyle{ \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]\displaystyle{ \sigma \gt 1, }[/math] [math]\displaystyle{ \sigma =1, }[/math] and [math]\displaystyle{ \sigma \lt 1 }[/math] ( [math]\displaystyle{ \beta \lt 1, }[/math] [math]\displaystyle{ \beta =1, }[/math] and [math]\displaystyle{ \beta \gt 1 }[/math] respectively).
• The exponential distribution is a special case when [math]\displaystyle{ \lambda =1 }[/math] and [math]\displaystyle{ \sigma =1 }[/math].
• The lognormal distribution is a special case when [math]\displaystyle{ \lambda =0 }[/math].
• The gamma distribution is a special case when [math]\displaystyle{ \lambda =\sigma }[/math].

By allowing [math]\displaystyle{ \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]\displaystyle{ \lambda =-1 }[/math].