Template:GeneralizedGammaDistribution

The Generalized Gamma Distribution

Compared to the other distributions previously discussed, the generalized gamma distribution is not as frequently used for modeling life data; however, it has the the ability to mimic the attributes of other distributions, such as the Weibull or lognormal, based on the values of the distribution’s parameters. This offers a compromise between two lifetime distributions. The generalized gamma function is a three-parameter distribution with parameters μ , σ and λ . The pdf of the distribution is given by,

[math]\displaystyle{ 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]

where Γ(x) is the gamma function, defined by:

[math]\displaystyle{ \Gamma (x)=\int_{0}^{\infty}{s}^{x-1}{e^{-s}}ds }[/math]

This distribution behaves as do other distributions based on the values of the parameters. For example, if λ = 1, then the distribution is identical to the Weibull distribution. If both λ = 1 and σ = 1, then the distribution is identical to the exponential distribution, and for λ = 0, 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