Generalized gamma Parameters
a
>
0
,
d
>
0
,
p
>
0
{\displaystyle a>0,d>0,p>0}
Support
x
∈
(
0
,
∞
)
{\displaystyle x\;\in \;(0,\,\infty )}
PDF
p
/
a
d
Γ
(
d
/
p
)
x
d
−
1
e
−
(
x
/
a
)
p
{\displaystyle {\frac {p/a^{d}}{\Gamma (d/p)}}x^{d-1}e^{-(x/a)^{p}}}
CDF
γ
(
d
/
p
,
(
x
/
a
)
p
)
Γ
(
d
/
p
)
{\displaystyle {\frac {\gamma (d/p,(x/a)^{p})}{\Gamma (d/p)}}}
Mean
a
Γ
(
(
d
+
1
)
/
p
)
Γ
(
d
/
p
)
{\displaystyle a{\frac {\Gamma ((d+1)/p)}{\Gamma (d/p)}}}
Mode
a
(
d
−
1
p
)
1
p
,
f
o
r
d
>
1
{\displaystyle a\left({\frac {d-1}{p}}\right)^{\frac {1}{p}},\mathrm {for} \;d>1}
Variance
a
2
(
Γ
(
(
d
+
2
)
/
p
)
Γ
(
d
/
p
)
−
(
Γ
(
(
d
+
1
)
/
p
)
Γ
(
d
/
p
)
)
2
)
{\displaystyle 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
a
Γ
(
d
/
p
)
p
+
d
p
+
(
1
p
−
d
p
)
ψ
(
d
p
)
{\displaystyle \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
{\displaystyle a>0}
,
d
>
0
{\displaystyle d>0}
, and
p
>
0
{\displaystyle p>0}
. For non-negative x , the probability density function of the generalized gamma is[2]
f
(
x
;
a
,
d
,
p
)
=
(
p
/
a
d
)
x
d
−
1
e
−
(
x
/
a
)
p
Γ
(
d
/
p
)
,
{\displaystyle f(x;a,d,p)={\frac {(p/a^{d})x^{d-1}e^{-(x/a)^{p}}}{\Gamma (d/p)}},}
where
Γ
(
⋅
)
{\displaystyle \Gamma (\cdot )}
denotes the gamma function .
The cumulative distribution function is
F
(
x
;
a
,
d
,
p
)
=
γ
(
d
/
p
,
(
x
/
a
)
p
)
Γ
(
d
/
p
)
,
{\displaystyle F(x;a,d,p)={\frac {\gamma (d/p,(x/a)^{p})}{\Gamma (d/p)}},}
where
γ
(
⋅
)
{\displaystyle \gamma (\cdot )}
denotes the lower incomplete gamma function .
If
d
=
p
{\displaystyle d=p}
then the generalized gamma distribution becomes the Weibull distribution . Alternatively, if
p
=
1
{\displaystyle 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]
E
(
X
r
)
=
a
r
Γ
(
d
+
r
p
)
Γ
(
d
p
)
.
{\displaystyle \operatorname {E} (X^{r})=a^{r}{\frac {\Gamma ({\frac {d+r}{p}})}{\Gamma ({\frac {d}{p}})}}.}
Kullback-Leibler divergence
If
f
1
{\displaystyle f_{1}}
and
f
2
{\displaystyle f_{2}}
are the probability density functions of two generalized gamma distributions, then their Kullback-Leibler divergence is given by
D
K
L
(
f
1
∥
f
2
)
=
∫
0
∞
f
1
(
x
;
a
1
,
d
1
,
p
1
)
ln
f
1
(
x
;
a
1
,
d
1
,
p
1
)
f
2
(
x
;
a
2
,
d
2
,
p
2
)
d
x
=
ln
p
1
a
2
d
2
Γ
(
d
2
/
p
2
)
p
2
a
1
d
1
Γ
(
d
1
/
p
1
)
+
[
ψ
(
d
1
/
p
1
)
p
1
+
ln
a
1
]
(
d
1
−
d
2
)
+
Γ
(
(
d
1
+
p
2
)
/
p
1
)
Γ
(
d
1
/
p
1
)
(
a
1
a
2
)
p
2
−
d
1
p
1
{\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
ψ
(
⋅
)
{\displaystyle \psi (\cdot )}
is the digamma function .[5]
Software implementation
In R implemented in the package flexsurv , function dgengamma , with different parametrisation:
μ
=
ln
a
+
ln
d
−
ln
p
p
{\displaystyle \mu =\ln a+{\frac {\ln d-\ln p}{p}}}
,
σ
=
1
p
d
{\displaystyle \sigma ={\frac {1}{\sqrt {pd}}}}
,
Q
=
p
d
{\displaystyle Q={\sqrt {\frac {p}{d}}}}
.
See also
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
^ a b c 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 .
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families