Generalized gamma distribution

The generalized gamma distribution has two shape parameters, $d\; >\; 0$ and $p\; >\; 0$, and a scale parameter, $a\; >\; 0$. For non-negative x from a generalized gamma distribution, the probability density function isStacy, E.W. (1962). "A Generalization of the Gamma Distribution." Annals of Mathematical Statistics 33(3): 1187-1192. {{JSTOR|2237889}}

:$$

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

where $\backslash Gamma(\backslash cdot)$ denotes the gamma function.

The cumulative distribution function is

:$$

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

P\left( \frac{d}{p}, \left( \frac{x}{a} \right)^p \right) ;

where $\backslash gamma(\backslash cdot)$ denotes the lower incomplete gamma function,

and $P(\backslash cdot,\; \backslash cdot)$ denotes the regularized lower incomplete gamma function.

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

:$$

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

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

• If $d=p$ then the generalized gamma distribution becomes the Weibull distribution.
• If $p=1$ the generalised gamma becomes the gamma distribution.
• If $p=d=1$ then it becomes the exponential distribution.
• If $p=d=2$ then it becomes the Rayleigh distribution.
• If $p=2$ and $d=2m$ then it becomes the Nakagami distribution.
• If $p=2$ and $d=1$ then it becomes a half-normal distribution.

Alternative parameterisations of this distribution are sometimes used; for example with the substitution α  =   d/p.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) In addition, a shift parameter can be added, so the domain of x starts at some value other than zero. 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.Gavin E. Crooks (2010), [https://threeplusone.com/pubs/on_amoroso.pdf The Amoroso Distribution], Technical Note, Lawrence Berkeley National Laboratory.

If X has a generalized gamma distribution as above, then

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

Denote GG(a,d,p) as the generalized gamma distribution of parameters a, d, p.

Then, given $c$ and $\backslash alpha$ two positive real numbers, if $f\; \backslash sim\; GG(a,d,p)$, then

$c\; f\backslash sim\; GG(c\; a,d,p)$ and

$f^\backslash alpha\backslash sim\; GG\backslash left(a^\backslash alpha,\backslash frac\{d\}\{\backslash alpha\},\backslash frac\{p\}\{\backslash alpha\}\backslash right)$.

If $f\_1$ and $f\_2$ are the probability density functions of two generalized gamma distributions, then their Kullback-Leibler divergence is given by

:$$

\begin{align}

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{align}

where $\backslash psi(\backslash cdot)$ is the digamma function.C. Bauckhage (2014), Computing the Kullback-Leibler Divergence between two Generalized Gamma Distributions, {{arXiv|1401.6853}}.

In the R programming language, there are a few packages that include functions for fitting and generating generalized gamma distributions. The [https://cran.r-project.org/web/packages/gamlss/index.html gamlss] package in R allows for fitting and generating many different distribution families including [https://rdrr.io/cran/gamlss.dist/man/GG.html generalized gamma] (family=GG). Other options in R, implemented in the package flexsurv, include the function dgengamma, with parameterization: $\backslash mu=\backslash ln\; a\; +\; \backslash frac\{\backslash ln\; d\; -\; \backslash ln\; p\}\{p\}$, $\backslash sigma=\backslash frac\{1\}\{\backslash sqrt\{pd\}\}$, $Q=\backslash sqrt\{\backslash frac\{p\}\{d\}\}$, and in the package ggamma with parametrisation: $a\; =\; a$, $b\; =\; p$, $k\; =\; d/p$.

In the python programming language, [https://docs.scipy.org/doc/scipy/tutorial/stats/continuous_gengamma.html it is implemented] in the SciPy package, with parametrisation: $c\; =\; p$, $a\; =\; d/p$, and scale of 1.

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

{{ProbDistributions|continuous-semi-infinite}}

Category:Continuous distributions

Category:Gamma and related functions