View source for Majumdar–Ghosh Model

{{Probability distribution 
| name       = Generalized gamma
| type       = density
| pdf_image  = 
| cdf_image  = 
| notation   = 
| parameters = <math>a>0, d>0, p>0</math>
| support    = <math>x \;\in\; (0,\, \infty)</math>
| pdf        = <math>\frac{p/a^d}{\Gamma(d/p)} x^{d-1}e^{-(x/a)^p}</math>
| cdf        = <math>\frac{\gamma(d/p, (x/a)^p)}{\Gamma(d/p)}</math>
| mean       = <math>a \frac{\Gamma((d+1)/p)}{\Gamma(d/p)}</math>
| median     = 
| mode       = <math>a \left(\frac{d-1}{p}\right)^{\frac{1}{p}}, \mathrm{for}\; d>1</math>
| variance   = <math>a^2\left(\frac{\Gamma((d+2)/p)}{\Gamma(d/p)} - \left(\frac{\Gamma((d+1)/p)}{\Gamma(d/p)}\right)^2\right)</math>
| skewness   = 
| kurtosis   = 
| entropy    = <math>\ln \frac{a \Gamma(d/p)}{p} + \frac{d}{p} + \left(\frac{1}{a}-\frac{d}{p}\right)\psi\left(\frac{d}{p}\right)</math>
| mgf        = 
| cf         = 
| pgf        =
| fisher     = 
}}
The '''generalized gamma distribution''' is a [[Continuous probability distribution|continuous]] [[probability distribution]] with three parameters. It is a generalization of the two-parameter [[gamma distribution]]. Since many distributions commonly used for parametric models in [[survival analysis]] (such as the [[Weibull distribution]] and the [[log-normal distribution]]) are special cases of the generalized gamma, it is sometimes used to determine which parametric model is appropriate for a given set of data.<ref>Box-Steffensmeier, Janet M.; Jones, Bradford S. (2004) ''Event History Modeling: A Guide for Social Scientists''. Cambridge University Press. ISBN 0-521-54673-7 (pp. 41-43)</ref>

==Characteristics==
The generalized gamma has three parameters: <math>a > 0</math>, <math>d > 0</math>, and <math>p > 0</math>. For non-negative ''x'', the [[probability density function]] of the generalized gamma is<ref>Stacy, E.W. (1962). "A Generalization of the Gamma Distribution." ''Annals of Mathematical Statistics'' 33(3): 1187-1192. {{jstor|2237889}}</ref>
:<math>
f(x; a, d, p) = \frac{(p/a^d) x^{d-1} e^{-(x/a)^p}}{\Gamma(d/p)},
</math>
where <math>\Gamma</math> denotes the [[gamma function]].

The [[cumulative distribution function]] is

:<math>
F(x; a, d, p) = \frac{\gamma(d/p, (x/a)^p)}{\Gamma(d/p)} ,</math>
where <math>\gamma</math> denotes the [[incomplete gamma function|lower incomplete gamma function]].

If <math>d=p</math> then the generalized gamma distribution becomes the [[Weibull distribution]].  Alternatively, if <math>p=1</math> the generalised gamma becomes the [[gamma distribution]].

Alternative parameterisations of this distribution are sometimes used; for example with the substitution ''&alpha; &nbsp;= &nbsp; d/p''.<ref name=JK>Johnson, N.L.; Kotz, S; Balakrishnan, N. (1994) ''Continuous Univariate Distributions, Volume 1'', 2nd Edition. Wiley.  ISBN 0-471-58495-9 (Section 17.8.7)</ref> In addition, a shift parameter can be added, so the domain of ''x'' starts at some value other than zero.<ref name=JK/> If the restrictions on the signs of ''a'', ''d'' and ''p'' are also lifted (but &alpha; = ''d''/''p'' remains positive), this gives a distribution called the '''Amoroso distribution''', after the Italian mathematician and economist [[Luigi Amoroso]] who described it in 1925.<ref>Gavin E. Crooks (2010), [http://threeplusone.com/Crooks-Amoroso.pdf The Amoroso Distribution], Technical Note, Lawrence Berkeley National Laboratory</ref>

==Moments==
If ''X'' has a generalized gamma distribution as above, then<ref name=JK/>
:<math>\operatorname{E}(X^r)= a^r \frac{\Gamma (\frac{d+r}{p})}{\Gamma( \frac{d}{p})} . </math>

==See also==
* [[Generalized integer gamma distribution]]

==References==
<references/>
{{ProbDistributions|continuous-semi-infinite}}
[[Category:Continuous distributions]]
[[Category:Probability distributions]]