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=== The Generalized Gamma Distribution  ===
#REDIRECT [[The_Generalized_Gamma_Distribution]]
 
Compared to the other distributions previously discussed, the generalized gamma distribution is not as frequently used for modeling life data;&nbsp;however, it has the&nbsp;the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution’s parameters.&nbsp;This&nbsp;offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters <span class="texhtml">μ</span> , <span class="texhtml">σ</span> and <span class="texhtml">λ</span> . The ''pdf ''of the distribution is given by,
 
::<math>
f(x)=\begin{cases}
\frac{|\lambda|}{\sigma \cdot t}\cdot \tfrac{1}{\Gamma( \tfrac{1}{\lambda}^2)}\cdot
{e^{\tfrac{\lambda \cdot{\tfrac{\ln(t)-\mu}{\sigma}}+\ln( \tfrac{1}{{\lambda}^2})-e^{\lambda \cdot {\tfrac{\ln(t)-\mu}{\sigma}}}}{{\lambda}^2}}}, & \text{if}  \ \lambda \ne 0 \\
 
\frac{1}{t\cdot \sigma \sqrt{2\pi }} e^{-\tfrac{1}{2}{(\tfrac{\ln(t)-\mu}{\sigma })^2}}, & \text{if} \ \lambda =0
\end{cases}
</math>
 
<br>where <span class="texhtml">Γ(''x'')</span> is the gamma function, defined by: <br>
 
::<math>\Gamma (x)=\int_{0}^{\infty}{s}^{x-1}{e^{-s}}ds</math>
 
<br>
 
This distribution behaves as do other distributions based on the values of the parameters. For example, if <span class="texhtml">λ = 1</span>, then the&nbsp;distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, then the&nbsp;distribution is identical to the exponential distribution, and for <span class="texhtml">λ = 0,</span> it is identical to the lognormal distribution. While the generalized gamma distribution is not often used to model life data by itself, its ability to behave like other more commonly-used life distributions is sometimes used to determine which of those life distributions should be used to model a particular set of data.
 
The Generalized Gamma distribution and its characteristics are presented in more detail in the chapter [[The Generalized Gamma Distribution]]  
 
<br>

Latest revision as of 09:33, 9 August 2012

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