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 Exponential distribution, the Weibull distribution and the Gamma 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{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 $\psi (\cdot )$ 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

List

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	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace 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	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-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 Dirac delta function Singular Cantor

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

This article is issued from Wikipedia - version of the 4/19/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.