Generalized gamma distribution

The generalized gamma has three parameters: ${\displaystyle a>0}$, ${\displaystyle d>0}$, and ${\displaystyle p>0}$. For non-negative x, the probability density function of the generalized gamma is[2]

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

where ${\displaystyle \Gamma (\cdot )}$ denotes the gamma function.

The cumulative distribution function is

${\displaystyle F(x;a,d,p)={\frac {\gamma (d/p,(x/a)^{p})}{\Gamma (d/p)}},}$

where ${\displaystyle \gamma (\cdot )}$ denotes the lower incomplete gamma function.

The quantile 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 about inverse of composite functions, yielding:

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

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 the Weibull distribution. Alternatively, if ${\displaystyle p=1}$ the generalised gamma becomes the gamma distribution. If ${\displaystyle p=2}$ and ${\displaystyle d=2m}$ then it becomes the Nakagami distribution.

Alternative parameterisations of this distribution are sometimes used; for example with the substitution α  =   d/p.[3] In addition, a shift parameter can be added, so the domain of x starts at some value other than zero.[3] If the restrictions on the signs of a, d and p are also lifted (but α = 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.[4]

If X has a generalized gamma distribution as above, then[3]

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

If ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}}$ are the probability density functions of two generalized gamma distributions, then their Kullback-Leibler divergence is given by

${\displaystyle {\begin{aligned}D_{KL}(f_{1}\parallel f_{2})&=\int _{0}^{\infty }f_{1}(x;a_{1},d_{1},p_{1})\,\ln {\frac {f_{1}(x;a_{1},d_{1},p_{1})}{f_{2}(x;a_{2},d_{2},p_{2})}}\,dx\\&=\ln {\frac {p_{1}\,a_{2}^{d_{2}}\,\Gamma \left(d_{2}/p_{2}\right)}{p_{2}\,a_{1}^{d_{1}}\,\Gamma \left(d_{1}/p_{1}\right)}}+\left[{\frac {\psi \left(d_{1}/p_{1}\right)}{p_{1}}}+\ln a_{1}\right](d_{1}-d_{2})+{\frac {\Gamma {\bigl (}(d_{1}+p_{2})/p_{1}{\bigr )}}{\Gamma \left(d_{1}/p_{1}\right)}}\left({\frac {a_{1}}{a_{2}}}\right)^{p_{2}}-{\frac {d_{1}}{p_{1}}}\end{aligned}}}$

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

In the R programming language, there are a few packages that include functions for fitting and generating generalized gamma distributions. The gamlss package in R allows for fitting and generating many different distribution families including generalized gamma (family=GG). Other options in R, implemented in the package flexsurv, include the function dgengamma, 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 package ggamma with parametrisation ${\displaystyle a=a}$, ${\displaystyle b=p}$, ${\displaystyle k=d/p}$.

• Generalized integer gamma distribution