Generalized gamma
Probability density function
Parameters	${\displaystyle a>0}$ (scale),${\displaystyle d,p>0}$
Support	${\displaystyle x\in (0,\mathrm{\infty })}$
PDF	${\displaystyle \frac{p/a^d}{\mathrm{\Gamma }(d/p)}{x}^{d-1}{e}^{-(x/a{)}^p}}$
CDF	${\displaystyle \frac{\gamma (d/p,(x/a{)}^p)}{\mathrm{\Gamma }(d/p)}}$
Mean	${\displaystyle a\frac{\mathrm{\Gamma }((d+1)/p)}{\mathrm{\Gamma }(d/p)}}$
Mode	${\displaystyle a{\left(\frac{d-1}{p}\right)}^{\frac{1}{p}}\mathrm{f}\mathrm{o}\mathrm{r}d>1,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}0}$
Variance	${\displaystyle a^2\left(\frac{\mathrm{\Gamma }((d+2)/p)}{\mathrm{\Gamma }(d/p)}-{\left(\frac{\mathrm{\Gamma }((d+1)/p)}{\mathrm{\Gamma }(d/p)}\right)}^2\right)}$
Entropy	${\displaystyle \ln \frac{a\mathrm{\Gamma }(d/p)}{p}+\frac{d}{p}+a\left(\frac{1}{p}-\frac{d}{p}\right)\psi \left(\frac{d}{p}\right)}$

Thegeneralized gamma distribution is acontinuousprobability distribution with twoshape parameters (and ascale parameter). It is a generalization of thegamma distribution which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models insurvival analysis (such as theexponential distribution, theWeibull distribution and thegamma 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.^[1] Another example is thehalf-normal distribution.

The generalized gamma distribution has twoshape parameters,${\displaystyle d>0}$ and${\displaystyle p>0}$, and ascale parameter,${\displaystyle a>0}$. For non-negativex from a generalized gamma distribution, theprobability density function is^[2]

${\displaystyle f(x;a,d,p)=\frac{(p/a^d)x^{d-1}{e}^{-(x/a{)}^p}}{\mathrm{\Gamma }(d/p)},}$

where${\displaystyle \mathrm{\Gamma }(\cdot )}$ denotes thegamma function.

Thecumulative distribution function is

${\displaystyle F(x;a,d,p)=\frac{\gamma (d/p,(x/a{)}^p)}{\mathrm{\Gamma }(d/p)},\text{or}P\left(\frac{d}{p},{\left(\frac{x}{a}\right)}^p\right);}$

where${\displaystyle \gamma (\cdot )}$ denotes thelower incomplete gamma function, and${\displaystyle P(\cdot ,\cdot )}$ denotes theregularized lower incomplete gamma function.

Thequantile function can be found by noting that${\displaystyle F(x;a,d,p)=G((x/a{)}^p)}$ where${\displaystyle G}$ is the cumulative distribution function of the gamma distribution with parameters${\displaystyle \alpha =d/p}$ and${\displaystyle \beta =1}$. The quantile function is then given by inverting${\displaystyle F}$ using known relations aboutinverse of composite functions, yielding:

${\displaystyle F^{-1}(q;a,d,p)=a\cdot [G^{-1}(q){]}^{1/p},}$

with${\displaystyle G^{-1}(q)}$ being the quantile function for a gamma distribution with${\displaystyle \alpha =d/p,\beta =1}$.

• If${\displaystyle d=p}$ then the generalized gamma distribution becomes theWeibull distribution.
• If${\displaystyle p=1}$ the generalised gamma becomes thegamma distribution.
• If${\displaystyle p=d=1}$ then it becomes theexponential distribution.
• If${\displaystyle p=d=2}$ then it becomes theRayleigh distribution.
• If${\displaystyle p=2}$ and${\displaystyle d=2m}$ then it becomes theNakagami distribution.
• If${\displaystyle p=2}$ and${\displaystyle d=1}$ then it becomes ahalf-normal distribution.

Alternative parameterisations of this distribution are sometimes used; for example with the substitutionα  =   d/p.^[3] In addition, a shift parameter can be added, so the domain ofx starts at some value other than zero.^[3] If the restrictions on the signs ofa,d andp are also lifted (but α =d/p remains positive), this gives a distribution called theAmoroso distribution, after the Italian mathematician and economistLuigi Amoroso who described it in 1925.^[4]

IfX has a generalized gamma distribution as above, then^[3]

${\displaystyle \mathrm{E}(X^r)=a^r\frac{\mathrm{\Gamma }(\frac{d+r}{p})}{\mathrm{\Gamma }(\frac{d}{p})}.}$

DenoteGG(a,d,p) as the generalized gamma distribution of parametersa,d,p.Then, given${\displaystyle c}$ and${\displaystyle \alpha }$ two positive real numbers, if${\displaystyle f\sim GG(a,d,p)}$, then${\displaystyle cf\sim GG(ca,d,p)}$ and${\displaystyle f^{\alpha }\sim GG\left(a^{\alpha },\frac{d}{\alpha },\frac{p}{\alpha }\right)}$.

If${\displaystyle f_1}$ and${\displaystyle f_2}$ are the probability density functions of two generalized gamma distributions, then theirKullback–Leibler divergence is given by

where${\displaystyle \psi (\cdot )}$ is thedigamma function.^[5]

In theR programming language, there are a few packages that include functions for fitting and generating generalized gamma distributions. Thegamlss package in R allows for fitting and generating many different distribution families includinggeneralized gamma (family=GG). Other options in R, implemented in the packageflexsurv, include the functiondgengamma, with parameterization:${\displaystyle \mu =\ln a+\frac{\ln d-\ln p}{p}}$,${\displaystyle \sigma =\frac{1}{\sqrt{pd}}}$,${\displaystyle Q=\sqrt{\frac{p}{d}}}$, and in the packageggamma with parametrisation:${\displaystyle a=a}$,${\displaystyle b=p}$,${\displaystyle k=d/p}$.

In thepython programming language,it is implemented in theSciPy package, with parametrisation:${\displaystyle c=p}$,${\displaystyle a=d/p}$, and scale of 1.

• Half-t distribution
• Truncated normal distribution
• Folded normal distribution
• Rectified Gaussian distribution
• Modified half-normal distribution
• Generalized integer gamma distribution

• Continuous distributions
• Gamma and related functions

• Articles with short description
• Short description matches Wikidata