# Generalized gamma distribution

Parameters $a>0, d>0, p>0$ $x \;\in\; (0,\, \infty)$ $\frac{p/a^d}{\Gamma(d/p)} x^{d-1}e^{-(x/a)^p}$ $\frac{\gamma(d/p, (x/a)^p)}{\Gamma(d/p)}$ $a \frac{\Gamma((d+1)/p)}{\Gamma(d/p)}$ $a \left(\frac{d-1}{p}\right)^{\frac{1}{p}}, \mathrm{for}\; d>1$ $a^2\left(\frac{\Gamma((d+2)/p)}{\Gamma(d/p)} - \left(\frac{\Gamma((d+1)/p)}{\Gamma(d/p)}\right)^2\right)$ $\ln \frac{a \Gamma(d/p)}{p} + \frac{d}{p} + \left(\frac{1}{a}-\frac{d}{p}\right)\psi\left(\frac{d}{p}\right)$

The generalized gamma distribution is a 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.[1]

## Characteristics

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

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

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

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

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

If $d=p$ then the generalized gamma distribution becomes the Weibull distribution. Alternatively, if $p=1$ the generalised gamma becomes the gamma 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]

## Moments

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

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

## Kullback-Leibler divergence

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 $\psi(\cdot)$ is the digamma function.[5]