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'''A Generalized Gamma Distribution Example'''
#REDIRECT [[Example:_Generalized_Gamma_Distribution_Example]]
[[Example: Generalized Gamma Distribution Example]]
The following data set represents revolutions-to-failure (in millions) for 23 ball bearings in a fatigue test [[Appendix: Weibull References|[21]]].
 
 
::<math>\begin{array}{*{35}{l}}
  \text{17}\text{.88} & \text{28}\text{.92} & \text{33} & \text{41}\text{.52} & \text{42}\text{.12} & \text{45}\text{.6} & \text{48}\text{.4} & \text{51}\text{.84} & \text{51}\text{.96} & \text{54}\text{.12}  \\
  \text{55}\text{.56} & \text{67}\text{.8} & \text{68}\text{.64} & \text{68}\text{.64} & \text{68}\text{.88} & \text{84}\text{.12} & \text{93}\text{.12} & \text{98}\text{.64} & \text{105}\text{.12} & \text{105}\text{.84}  \\
  \text{127}\text{.92} & \text{128}\text{.04} & \text{173}\text{.4} & {} & {} & {} & {} & {} & {} & {}  \\
\end{array}</math>
 
 
When the generalized gamma distribution is fitted to this data using MLE, the following values for parameters are obtained:
 
::<math>\begin{align}
  & \widehat{\mu }= & 4.23064 \\
& \widehat{\sigma }= & 0.509982 \\
& \widehat{\lambda }= & 0.307639 
\end{align}</math>
 
Note that for this data, the generalized gamma offers a compromise between the Weibull  <math>(\lambda =1),</math>  and the lognormal  <math>(\lambda =0)</math>  distributions. The value of  <math>\lambda </math>  indicates that the lognormal distribution is better supported by the data. A better assessment, however, can be made by looking at the confidence bounds on  <math>\lambda .</math>  For example, the 90% two-sided confidence bounds are:
 
::<math>\begin{align}
  & {{\lambda }_{u}}= & -0.592087 \\
& {{\lambda }_{u}}= & 1.20736 
\end{align}</math>
 
It can be then concluded that both distributions (i.e. Weibull and lognormal) are well supported by the data, with the lognormal being the ,better supported of the two.
In Weibull++ the generalized gamma probability is plotted on gamma probability paper, as shown next.
 
[[Image:GGamma Example 1 Plot.png|thumb|center|500px| ]]  
 
It is important to also note that as in the case of the mixed Weibull distribution, in the case of regression analysis, using a generalized gamma model, the choice of regression axis, i.e.  <math>RRX</math>  or  <math>RRY,</math>  is of no consequence since non-linear regression is utilized.

Latest revision as of 03:58, 15 August 2012

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