Generalized gamma distribution

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

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.

The cumulative distribution function is

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]

Software implementation

In R implemented in the package flexsurv, function dgengamma, with different parametrisation: $\mu=\ln a + \frac{\ln d - \ln p}{p}$ , $\sigma=\frac{1}{\sqrt{pd}}$ , $Q=\sqrt{\frac{p}{d}}$ .

References

↑ 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)
↑ Stacy, E.W. (1962). "A Generalization of the Gamma Distribution." Annals of Mathematical Statistics 33(3): 1187-1192. JSTOR 2237889
1 2 3 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)
↑ Gavin E. Crooks (2010), The Amoroso Distribution, Technical Note, Lawrence Berkeley National Laboratory.
↑ C. Bauckhage (2014), Computing the Kullback-Leibler Divergence between two Generalized Gamma Distributions, arXiv:1401.6853.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace Asymmetric Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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