Gamma/Gompertz distribution
Gamma/Gompertz distribution
Probability density function
Note: b=0.4, β=3
Cumulative distribution function
Parameters
b
,
s
,
β β -->
>
0
{\displaystyle b,s,\beta >0\,\!}
Support
x
∈ ∈ -->
[
0
,
∞ ∞ -->
)
{\displaystyle x\in [0,\infty )\!}
PDF
b
s
e
b
x
β β -->
s
/
(
β β -->
− − -->
1
+
e
b
x
)
s
+
1
where
b
,
s
,
β β -->
>
0
{\displaystyle bse^{bx}\beta ^{s}/\left(\beta -1+e^{bx}\right)^{s+1}{\text{where }}b,s,\beta >0}
CDF
1
− − -->
β β -->
s
/
(
β β -->
− − -->
1
+
e
b
x
)
s
,
x
>
0
,
b
,
s
,
β β -->
>
0
{\displaystyle 1-\beta ^{s}/\left(\beta -1+e^{bx}\right)^{s},x>0,b,s,\beta >0}
1
− − -->
e
− − -->
b
s
x
,
β β -->
=
1
{\displaystyle 1-e^{-bsx},\beta =1}
Mean
=
(
1
/
b
)
(
1
/
s
)
2
F
1
(
s
,
1
;
s
+
1
;
(
β β -->
− − -->
1
)
/
β β -->
)
,
{\displaystyle =\left(1/b\right)\left(1/s\right){_{2}{\text{F}}_{1}}\left(s,1;s+1;\left(\beta -1\right)/\beta \right),}
b
,
s
>
0
,
β β -->
≠ ≠ -->
1
{\displaystyle b,s>0,\beta \neq 1}
=
(
1
/
b
)
[
β β -->
/
(
β β -->
− − -->
1
)
]
ln
-->
(
β β -->
)
,
{\displaystyle =\left(1/b\right)\left[\beta /\left(\beta -1\right)\right]\ln \left(\beta \right),}
b
>
0
,
s
=
1
,
β β -->
≠ ≠ -->
1
{\displaystyle b>0,s=1,\beta \neq 1}
=
1
/
(
b
s
)
,
b
,
s
>
0
,
β β -->
=
1
{\displaystyle =1/\left(bs\right),\quad b,s>0,\beta =1}
Median
(
1
/
b
)
ln
-->
{
β β -->
[
(
1
/
2
)
− − -->
1
/
s
− − -->
1
]
+
1
}
{\displaystyle \left(1/b\right)\ln\{\beta \left[\left(1/2\right)^{-1/s}-1\right]+1\}}
Mode
x
∗ ∗ -->
=
(
1
/
b
)
ln
-->
[
(
1
/
s
)
(
β β -->
− − -->
1
)
]
,
with
0
<
F
(
x
∗ ∗ -->
)
<
1
− − -->
(
β β -->
s
)
s
/
[
(
β β -->
− − -->
1
)
(
s
+
1
)
]
s
<
0.632121
,
β β -->
>
s
+
1
=
0
,
β β -->
≤ ≤ -->
s
+
1
{\displaystyle {\begin{aligned}x^{*}&=(1/b)\ln \left[(1/s)(\beta -1)\right],\\&{\text{with }}0<{\text{F}}(x^{*})<1-(\beta s)^{s}/\left[(\beta -1)(s+1)\right]^{s}<0.632121,\\&\beta >s+1\\&=0,\quad \beta \leq s+1\\\end{aligned}}}
Variance
=
2
(
1
/
b
2
)
(
1
/
s
2
)
β β -->
s
3
F
2
(
s
,
s
,
s
;
s
+
1
,
s
+
1
;
1
− − -->
β β -->
)
{\displaystyle =2(1/b^{2})(1/s^{2})\beta ^{s}{_{3}{\text{F}}_{2}}(s,s,s;s+1,s+1;1-\beta )}
− − -->
E
2
(
τ τ -->
|
b
,
s
,
β β -->
)
,
β β -->
≠ ≠ -->
1
{\displaystyle -{\text{E}}^{2}(\tau |b,s,\beta ),\quad \beta \neq 1}
=
(
1
/
b
2
)
(
1
/
s
2
)
,
β β -->
=
1
{\displaystyle =(1/b^{2})(1/s^{2}),\quad \beta =1}
with
{\displaystyle {\text{with}}}
3
F
2
(
a
,
b
,
c
;
d
,
e
;
z
)
=
∑ ∑ -->
k
=
0
∞ ∞ -->
{
(
a
)
k
(
b
)
k
(
c
)
k
/
[
(
d
)
k
(
e
)
k
]
}
z
k
/
k
!
{\displaystyle {_{3}{\text{F}}_{2}}(a,b,c;d,e;z)=\sum _{k=0}^{\infty }\{(a)_{k}(b)_{k}(c)_{k}/[(d)_{k}(e)_{k}]\}z^{k}/k!}
and
{\displaystyle {\text{and}}}
(
a
)
k
=
Γ Γ -->
(
a
+
k
)
/
Γ Γ -->
(
a
)
{\displaystyle (a)_{k}=\Gamma (a+k)/\Gamma (a)}
MGF
E
(
e
− − -->
t
x
)
{\displaystyle {\text{E}}(e^{-tx})}
=
β β -->
s
[
s
b
/
(
t
+
s
b
)
]
2
F
1
(
s
+
1
,
(
t
/
b
)
+
s
;
(
t
/
b
)
+
s
+
1
;
1
− − -->
β β -->
)
,
{\displaystyle =\beta ^{s}[sb/(t+sb)]{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),}
β β -->
≠ ≠ -->
1
{\displaystyle \quad \beta \neq 1}
=
s
b
/
(
t
+
s
b
)
,
β β -->
=
1
{\displaystyle =sb/(t+sb),\quad \beta =1}
with
2
F
1
(
a
,
b
;
c
;
z
)
=
∑ ∑ -->
k
=
0
∞ ∞ -->
[
(
a
)
k
(
b
)
k
/
(
c
)
k
]
z
k
/
k
!
{\displaystyle {\text{with }}{_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!}
In probability and statistics , the Gamma/Gompertz distribution is a continuous probability distribution . It has been used as an aggregate-level model of customer lifetime and a model of mortality risks.
Specification
Probability density function
The probability density function of the Gamma/Gompertz distribution is:
f
(
x
;
b
,
s
,
β β -->
)
=
b
s
e
b
x
β β -->
s
(
β β -->
− − -->
1
+
e
b
x
)
s
+
1
{\displaystyle f(x;b,s,\beta )={\frac {bse^{bx}\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s+1}}}}
where
b
>
0
{\displaystyle b>0}
scale parameter and
β β -->
,
s
>
0
{\displaystyle \beta ,s>0\,\!}
shape parameters of the Gamma/Gompertz distribution.
Cumulative distribution function
The cumulative distribution function of the Gamma/Gompertz distribution is:
F
(
x
;
b
,
s
,
β β -->
)
=
1
− − -->
β β -->
s
(
β β -->
− − -->
1
+
e
b
x
)
s
,
x
>
0
,
b
,
s
,
β β -->
>
0
=
1
− − -->
e
− − -->
b
s
x
,
β β -->
=
1
{\displaystyle {\begin{aligned}F(x;b,s,\beta )&=1-{\frac {\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s}}},{\ }x>0,{\ }b,s,\beta >0\\[6pt]&=1-e^{-bsx},{\ }\beta =1\\\end{aligned}}}
Moment generating function
The moment generating function is given by:
E
(
e
− − -->
t
x
)
=
{
β β -->
s
s
b
t
+
s
b
2
F
1
(
s
+
1
,
(
t
/
b
)
+
s
;
(
t
/
b
)
+
s
+
1
;
1
− − -->
β β -->
)
,
β β -->
≠ ≠ -->
1
;
s
b
t
+
s
b
,
β β -->
=
1.
{\displaystyle {\begin{aligned}{\text{E}}(e^{-tx})={\begin{cases}\displaystyle \beta ^{s}{\frac {sb}{t+sb}}{\ }{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),&\beta \neq 1;\\\displaystyle {\frac {sb}{t+sb}},&\beta =1.\end{cases}}\end{aligned}}}
where
2
F
1
(
a
,
b
;
c
;
z
)
=
∑ ∑ -->
k
=
0
∞ ∞ -->
[
(
a
)
k
(
b
)
k
/
(
c
)
k
]
z
k
/
k
!
{\displaystyle {_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!}
Hypergeometric function .
Properties
The Gamma/Gompertz distribution is a flexible distribution that can be skewed to the right or to the left.
When β = 1, this reduces to an Exponential distribution with parameter sb .
The gamma distribution is a natural conjugate prior to a Gompertz likelihood with known, scale parameter
b
.
{\displaystyle b\,\!.}
[ 1]
When the shape parameter
η η -->
{\displaystyle \eta \,\!}
Gompertz distribution varies according to a gamma distribution with shape parameter
α α -->
{\displaystyle \alpha \,\!}
β β -->
{\displaystyle \beta \,\!}
α α -->
/
β β -->
{\displaystyle \alpha /\beta \,\!}
x
{\displaystyle x}
[ 1]
See also
Notes
^ a b Bemmaor, A.C.; Glady, N. (2012)
References
Bemmaor, Albert C.; Glady, Nicolas (2012). "Modeling Purchasing Behavior With Sudden 'Death': A Flexible Customer Lifetime Model" . Management Science . 58 (5): 1012– 1021. doi :10.1287/mnsc.1110.1461 . Archived from the original on 2015-06-26. Bemmaor, Albert C.; Glady, Nicolas (2011). "Implementing the Gamma/Gompertz/NBD Model in MATLAB" (PDF) . Cergy-Pontoise: ESSEC Business School. [permanent dead link ] Gompertz, B. (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies" . Philosophical Transactions of the Royal Society of London 115 : 513– 583. doi :10.1098/rstl.1825.0026 JSTOR 107756 . S2CID 145157003 . Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1995). Continuous Univariate Distributions . Vol. 2 (2nd ed.). New York: John Wiley & Sons. pp. 25– 26. ISBN 0-471-58494-0 Manton, K. G.; Stallard, E.; Vaupel, J. W. (1986). "Alternative Models for the Heterogeneity of Mortality Risks Among the Aged". Journal of the American Statistical Association 81 (395): 635– 644. doi :10.1080/01621459.1986.10478316 . PMID 12155405 .
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\displaystyle =\left(1/b\right)\left[\beta /\left(\beta -1\right)\right]\ln \left(\beta \right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d40ede0194fe8297e30eb62598a07b1aaebf91)
![{\displaystyle \left(1/b\right)\ln\{\beta \left[\left(1/2\right)^{-1/s}-1\right]+1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20459f70785a4f564b5dbb1ee080f91070add07c)
![{\displaystyle {\begin{aligned}x^{*}&=(1/b)\ln \left[(1/s)(\beta -1)\right],\\&{\text{with }}0<{\text{F}}(x^{*})<1-(\beta s)^{s}/\left[(\beta -1)(s+1)\right]^{s}<0.632121,\\&\beta >s+1\\&=0,\quad \beta \leq s+1\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c310906d01a8b2136c968ee71c67056320464d7)
![{\displaystyle {_{3}{\text{F}}_{2}}(a,b,c;d,e;z)=\sum _{k=0}^{\infty }\{(a)_{k}(b)_{k}(c)_{k}/[(d)_{k}(e)_{k}]\}z^{k}/k!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9de7340c6d52753050a5ae813e06912c61335448)
![{\displaystyle =\beta ^{s}[sb/(t+sb)]{_{2}{\text{F}}_{1}}(s+1,(t/b)+s;(t/b)+s+1;1-\beta ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ec718073cb46360bb3076cc6daef007b2da181)
![{\displaystyle {\text{with }}{_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbecfda266daf8e2d14e0749ef8a6b773b5be24)
![{\displaystyle {\begin{aligned}F(x;b,s,\beta )&=1-{\frac {\beta ^{s}}{\left(\beta -1+e^{bx}\right)^{s}}},{\ }x>0,{\ }b,s,\beta >0\\[6pt]&=1-e^{-bsx},{\ }\beta =1\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0500f204c3a87c90d1263b07146dd09eb7d199)
![{\displaystyle {_{2}{\text{F}}_{1}}(a,b;c;z)=\sum _{k=0}^{\infty }[(a)_{k}(b)_{k}/(c)_{k}]z^{k}/k!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6944910aa72c697ec5ee711df72ca472e877fef5)
