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=== The Generalized Gamma Distribution  ===
#REDIRECT [[The_Generalized_Gamma_Distribution]]
While not as frequently used for modeling life data as the distributions discussed previously, the generalized gamma distribution does have the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution’s parameters and also offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters <span class="texhtml">μ</span> , <math>\sigma</math> 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>, the distribution is identical to the Weibull distribution. If both <span class="texhtml">λ = 1</span> and <span class="texhtml">σ = 1</span>, the 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.
 
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Latest revision as of 09:33, 9 August 2012

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