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Declined by Chaotic Enby Last edited by Chaotic Enby 56 days ago. Reviewer: Inform author . Resubmit Please note that if the issues are not fixed, the draft will be declined again.
Probability distribution
Xgamma distribution Parameters
θ
>
0
,
{\displaystyle \theta >0,}
shape Support
x
∈
[
0
,
∞
)
{\displaystyle x\in [0,\infty )}
PDF
θ
2
(
1
+
θ
)
(
1
+
θ
2
x
2
)
e
−
θ
x
{\displaystyle {\frac {\theta ^{2}}{(1+\theta )}}\left(1+{\frac {\theta }{2}}{x^{2}}\right)e^{-\theta x}}
CDF
1
−
(
1
+
θ
+
θ
x
+
θ
2
x
2
2
)
(
1
+
θ
)
e
−
θ
x
{\displaystyle 1-{\frac {(1+\theta +\theta x+{\frac {\theta ^{2}x^{2}}{2}})}{(1+\theta )}}e^{-\theta x}}
Mean
(
θ
+
3
)
θ
(
1
+
θ
)
{\displaystyle {\frac {(\theta +3)}{\theta (1+\theta )}}}
Mode
1
+
1
−
2
θ
θ
,
for
θ
≤
1
/
2
{\displaystyle {\frac {1+{\sqrt {1-2\theta }}}{\theta }},{\text{ for }}\theta \leq 1/2}
Variance
(
θ
2
+
8
θ
+
3
)
θ
2
(
1
+
θ
)
2
{\displaystyle {\frac {(\theta ^{2}+8\theta +3)}{\theta ^{2}(1+\theta )^{2}}}}
Skewness
2
(
θ
3
+
15
θ
2
+
9
θ
+
3
)
(
θ
2
+
8
θ
+
3
)
3
/
2
{\displaystyle {\frac {2(\theta ^{3}+15\theta ^{2}+9\theta +3)}{(\theta ^{2}+8\theta +3)^{3/2}}}}
MGF
θ
2
(
1
+
θ
)
[
1
(
θ
−
t
)
+
θ
(
θ
−
t
)
3
]
{\displaystyle {\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -t)}}+{\frac {\theta }{(\theta -t)^{3}}}\right]}
CF
θ
2
(
1
+
θ
)
[
1
(
θ
−
i
t
)
+
θ
(
θ
−
i
t
)
3
]
{\displaystyle {\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -it)}}+{\frac {\theta }{(\theta -it)^{3}}}\right]}
In probability theory and statistics , the xgamma distribution is continuous probability distribution (introduced by Sen et al. in 2016 [1] ). This distribution is obtained as a special finite mixture of exponentia l and gamma distributions. This distribution is successfully used in modelling time-to-event or lifetime data sets coming from diverse fields.
Exponential distribution with parameter θ and gamma distribution with scale parameter θ and shape parameter 3 are mixed mixing proportions,
θ
(
1
+
θ
)
{\displaystyle {\frac {\theta }{(1+\theta )}}}
1
(
1
+
θ
)
{\displaystyle {\frac {1}{(1+\theta )}}}
Probability density function
edit
The probability density function (pdf) of an xgamma distribution is[ 1]
f
(
x
;
θ
)
=
{
θ
2
(
1
+
θ
)
(
1
+
θ
2
x
2
)
e
−
θ
x
x
≥
0
,
0
x
<
0.
{\displaystyle f(x;\theta )={\begin{cases}{\frac {\theta ^{2}}{(1+\theta )}}\left(1+{\frac {\theta }{2}}{x^{2}}\right)e^{-\theta x}&x\geq 0,\\0&x<0.\end{cases}}}
Here θ > 0 is the parameter of the distribution, often called the shape parameter . The distribution is supported on the interval [0, ∞) . If a random variable X has this distribution, we write X ~ XG(θ )
Cumulative distribution function
edit
The cumulative distribution function is given by
F
(
x
;
θ
)
=
{
1
−
(
1
+
θ
+
θ
x
+
θ
2
x
2
2
)
(
1
+
θ
)
e
−
θ
x
x
≥
0
,
0
x
<
0.
{\displaystyle F(x;\theta )={\begin{cases}1-{\frac {(1+\theta +\theta x+{\frac {\theta ^{2}x^{2}}{2}})}{(1+\theta )}}e^{-\theta x}&x\geq 0,\\0&x<0.\end{cases}}}
Characteristic and generating functions
edit
The characteristic function of a random variable following xgamma distribution with parameter θ is given by[ 2]
ϕ
X
(
t
)
=
E
[
e
i
t
X
]
=
θ
2
(
1
+
θ
)
[
1
(
θ
−
i
t
)
+
θ
(
θ
−
i
t
)
3
]
;
t
∈
R
,
i
=
−
1
.
{\displaystyle \phi _{X}(t)=E[e^{itX}]={\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -it)}}+{\frac {\theta }{(\theta -it)^{3}}}\right];t\in \mathbb {R} ,i={\sqrt {-1}}.}
The moment generating function of xgamma distribution is given by
M
X
(
t
)
=
E
[
e
t
X
]
=
θ
2
(
1
+
θ
)
[
1
(
θ
−
t
)
+
θ
(
θ
−
t
)
3
]
;
t
∈
R
.
{\displaystyle M_{X}(t)=E[e^{tX}]={\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -t)}}+{\frac {\theta }{(\theta -t)^{3}}}\right];t\in \mathbb {R} .}
edit
The non-central moments of X , for
r
∈
N
{\displaystyle r\in \mathbb {N} }
μ
r
′
=
r
!
[
2
θ
+
(
r
+
1
)
(
r
+
2
)
]
2
θ
r
(
1
+
θ
)
.
{\displaystyle \mu _{r}'={\frac {r![2\theta +(r+1)(r+2)]}{2\theta ^{r}(1+\theta )}}.}
In particular,
The mean or expected value of a random variable X following xgamma distribution with parameter θ is given by
E
[
X
]
=
(
θ
+
3
)
θ
(
1
+
θ
)
.
{\displaystyle \operatorname {E} [X]={\frac {(\theta +3)}{\theta (1+\theta )}}.}
The
r
t
h
{\displaystyle r^{th}}
(
r
∈
N
)
{\displaystyle (r\in \mathbb {N} )}
μ
r
=
E
[
(
X
−
μ
)
r
]
=
∑
j
=
0
r
(
r
j
)
μ
r
′
(
−
μ
)
r
−
j
,
{\displaystyle \mu _{r}=E[{(X-\mu )}^{r}]=\sum _{j=0}^{r}{\binom {r}{j}}\mu _{r}{'}(-{\mu })^{r-j},}
μ
{\displaystyle \mu }
The variance of X is given by
Var
[
X
]
=
(
θ
2
+
8
θ
+
3
)
θ
2
(
1
+
θ
)
2
.
{\displaystyle \operatorname {Var} [X]={\frac {(\theta ^{2}+8\theta +3)}{\theta ^{2}(1+\theta )^{2}}}.}
The mode of xgamma distribution is given by
M
o
d
e
[
X
]
=
{
1
+
1
−
2
θ
θ
if
0
<
θ
≤
1
/
2
,
0
otherwise
.
{\displaystyle Mode[X]={\begin{cases}{\frac {1+{\sqrt {1-2\theta }}}{\theta }}&{\text{if}}&0<\theta \leq 1/2,\\0&{\text{otherwise}}.\end{cases}}}
Skewness and kurtosis
edit
The coefficients of skewness and kurtosis of xgamma distribution with parameter θ show that the distribution is positively skewed.
Measure of skewness:
β
1
=
μ
3
2
μ
2
3
=
2
(
θ
3
+
15
θ
2
+
9
θ
+
3
)
(
θ
2
+
8
θ
+
3
)
3
/
2
.
{\displaystyle {\sqrt {\beta _{1}}}={\sqrt {\frac {\mu _{3}^{2}}{\mu _{2}^{3}}}}={\frac {2(\theta ^{3}+15\theta ^{2}+9\theta +3)}{(\theta ^{2}+8\theta +3)^{3/2}}}.}
Measure of kurtosis:
β
2
=
μ
4
μ
2
2
=
3
(
5
θ
4
+
88
θ
3
+
310
θ
2
+
288
θ
+
177
)
(
θ
2
+
8
θ
+
3
)
2
.
{\displaystyle \beta _{2}={\frac {\mu _{4}}{\mu _{2}^{2}}}={\frac {3(5\theta ^{4}+88\theta ^{3}+310\theta ^{2}+288\theta +177)}{(\theta ^{2}+8\theta +3)^{2}}}.}
Among survival properties, failure rate or hazard rate function, mean residual life function and stochastic order relations are well established for xgamma distribution with parameter θ .
The survival function at time point t(> 0) is given by
S
(
t
;
θ
)
=
Pr
(
X
>
t
)
=
(
1
+
θ
+
θ
t
+
θ
2
t
2
2
)
(
1
+
θ
)
e
−
θ
t
.
{\displaystyle S(t;\theta )=\Pr(X>t)={\frac {(1+\theta +\theta t+{\frac {\theta ^{2}t^{2}}{2}})}{(1+\theta )}}e^{-\theta t}.}
Failure rate or Hazard rate function
edit
For xgamma distribution, the hazard rate (or failure rate) function is obtained as
h
(
t
;
θ
)
=
θ
2
(
1
+
θ
2
t
2
)
(
1
+
θ
+
θ
t
+
θ
2
2
t
2
)
.
{\displaystyle h(t;\theta )={\frac {\theta ^{2}(1+{\frac {\theta }{2}}t^{2})}{(1+\theta +\theta t+{\frac {\theta ^{2}}{2}}t^{2})}}.}
The hazard rate function in possesses the following properties.
lim
t
→
0
h
(
t
;
θ
)
=
θ
2
(
1
+
θ
)
=
lim
t
→
0
f
(
t
;
θ
)
.
{\displaystyle \lim _{t\to 0}h(t;\theta )={\frac {\theta ^{2}}{(1+\theta )}}=\lim _{t\to 0}f(t;\theta ).}
h
(
t
;
θ
)
{\displaystyle h(t;\theta )}
t
>
2
/
θ
.
{\displaystyle t>{\sqrt {2/\theta }}.}
θ
2
/
(
1
+
θ
)
<
h
(
t
;
θ
)
<
θ
.
{\displaystyle \theta ^{2}/(1+\theta )<h(t;\theta )<\theta .}
edit
Mean residual life (MRL) function related to a life time probability distribution is an important characteristic useful in survival analysis and reliability engineering .
The MRL function for xgamma distribution is given by
m
(
t
;
θ
)
=
1
θ
+
(
2
+
θ
t
)
θ
(
1
+
θ
+
θ
t
+
θ
2
2
t
2
)
.
{\displaystyle m(t;\theta )={\frac {1}{\theta }}+{\frac {(2+\theta t)}{\theta (1+\theta +\theta t+{\frac {\theta ^{2}}{2}}t^{2})}}.}
This MRL function has the following properties.
lim
t
→
0
m
(
t
;
θ
)
=
E
[
X
]
=
(
θ
+
3
)
θ
(
1
+
θ
)
{\displaystyle \lim _{t\to 0}m(t;\theta )=E[X]={\frac {(\theta +3)}{\theta (1+\theta )}}}
m
(
t
;
θ
)
{\displaystyle m(t;\theta )}
t and
θ
{\displaystyle \theta }
1
θ
<
m
(
t
;
θ
)
<
(
θ
+
3
)
θ
(
1
+
θ
)
.
{\displaystyle {\frac {1}{\theta }}<m(t;\theta )<{\frac {(\theta +3)}{\theta (1+\theta )}}.}
Statistical inference
edit
Below are provided two classical methods, namely maximum of estimation for the unknown parameter of xgamma distribution under complete sample set up.
Parameter estimation
edit
Maximum likelihood estimation
edit
Let
x
=
(
x
1
,
x
2
,
…
,
x
n
)
{\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})}
X
1
,
X
2
,
⋯
,
X
n
{\displaystyle X_{1},X_{2},\cdots ,X_{n}}
L
(
θ
|
x
)
=
∏
i
=
1
n
θ
2
(
1
+
θ
)
(
1
+
θ
2
x
i
2
)
e
−
θ
x
i
.
{\displaystyle L(\theta |x)=\prod _{i=1}^{n}{\frac {\theta ^{2}}{(1+\theta )}}\left(1+{\frac {\theta }{2}}x_{i}^{2}\right)e^{-\theta x_{i}}.}
l
(
θ
)
=
ln
L
(
θ
|
x
)
=
2
n
ln
θ
−
n
ln
(
1
+
θ
)
+
∑
i
=
1
n
ln
(
1
+
θ
2
x
i
2
)
−
θ
∑
i
=
1
n
x
i
.
{\displaystyle l(\theta )=\ln {L(\theta |x)}=2n\ln {\theta }-n\ln(1+\theta )+\sum _{i=1}^{n}\ln {\left(1+{\frac {\theta }{2}}x_{i}^{2}\right)}-\theta \sum _{i=1}^{n}x_{i}.}
To obtain maximum likelihood estimator (MLE) of
θ
{\displaystyle \theta }
θ
^
{\displaystyle {\hat {\theta }}}
θ
{\displaystyle \theta }
∂
ln
L
(
θ
|
x
)
∂
θ
=
0.
{\displaystyle {\frac {\partial \ln {L(\theta |x)}}{\partial \theta }}=0.}
∂
ln
L
(
θ
|
x
)
∂
θ
=
0
{\displaystyle {\frac {\partial \ln {L(\theta |x)}}{\partial \theta }}=0}
Newton-Raphson algorithm is applied to solve.
The initial solution for such an iteration can be taken as
θ
0
=
n
∑
i
=
1
n
x
i
.
{\displaystyle \theta _{0}={\frac {n}{\sum _{i=1}^{n}x_{i}}}.}
θ
(
i
)
=
θ
(
i
−
1
)
−
l
(
θ
(
i
−
1
)
|
x
)
l
′
(
θ
(
i
−
1
)
|
x
)
{\displaystyle \theta ^{(i)}=\theta ^{(i-1)}-{\frac {l(\theta ^{(i-1)}|x)}{l^{'}(\theta ^{(i-1)}|x)}}}
θ
(
i
)
{\displaystyle \theta ^{(i)}}
θ
(
i
)
≅
θ
(
i
−
1
)
{\displaystyle \theta ^{(i)}\cong \theta ^{(i-1)}}
Method of moments estimation
edit
Given a random sample
X
1
,
X
2
,
…
,
X
n
{\displaystyle X_{1},X_{2},\ldots ,X_{n}}
n from the xgamma distribution, the moment estimator for the parameter
θ
{\displaystyle \theta }
X
¯
=
1
n
∑
i
=
1
n
X
i
{\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}
X
¯
=
(
θ
+
3
)
θ
(
1
+
θ
)
,
{\displaystyle {\bar {X}}={\frac {(\theta +3)}{\theta (1+\theta )}},}
θ
{\displaystyle \theta }
X
¯
θ
2
+
(
X
¯
−
1
)
θ
−
3
=
0.
{\displaystyle {\bar {X}}\theta ^{2}+({\bar {X}}-1)\theta -3=0.}
θ
M
^
{\displaystyle {\hat {\theta _{M}}}}
θ
{\displaystyle \theta }
θ
^
M
=
−
(
X
¯
−
1
)
+
(
X
¯
−
1
)
2
+
12
X
¯
2
X
¯
for
X
¯
>
0.
{\displaystyle {\hat {\theta }}_{M}={\frac {-({\bar {X}}-1)+{\sqrt {({\bar {X}}-1)^{2}+12{\bar {X}}}}}{2{\bar {X}}}}\;{\text{for}}\;{\bar {X}}>0.}
Random variate generation
edit
To generate random data
X
i
;
i
=
1
,
2
,
…
,
n
,
{\displaystyle X_{i};i=1,2,\ldots ,n,}
θ
{\displaystyle \theta }
Generate
U
i
∼
u
n
i
f
o
r
m
(
0
,
1
)
,
i
=
1
,
2
,
…
,
n
.
{\displaystyle U_{i}\sim uniform(0,1),i=1,2,\ldots ,n.}
Generate
V
i
∼
e
x
p
(
θ
)
,
i
=
1
,
2
,
…
,
n
.
{\displaystyle V_{i}\sim exp(\theta ),i=1,2,\ldots ,n.}
Generate
W
i
∼
g
a
m
m
a
(
3
,
θ
)
,
i
=
1
,
2
,
…
,
n
.
{\displaystyle W_{i}\sim gamma(3,\theta ),i=1,2,\ldots ,n.}
If
U
i
≤
θ
/
(
1
+
θ
)
{\displaystyle U_{i}\leq \theta /(1+\theta )}
X
i
=
V
i
{\displaystyle X_{i}=V_{i}}
X
i
=
W
i
.
{\displaystyle X_{i}=W_{i}.}
![{\displaystyle {\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -t)}}+{\frac {\theta }{(\theta -t)^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e59bf9d48e9c1e50bafe91cd05afea50ab13ef0e)
![{\displaystyle {\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -it)}}+{\frac {\theta }{(\theta -it)^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3774e1da6b3328965b726258d65f623002df44c)
![{\displaystyle \phi _{X}(t)=E[e^{itX}]={\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -it)}}+{\frac {\theta }{(\theta -it)^{3}}}\right];t\in \mathbb {R} ,i={\sqrt {-1}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25c7a2e4607557fcf230130766a15ce6ce611ffe)
![{\displaystyle M_{X}(t)=E[e^{tX}]={\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -t)}}+{\frac {\theta }{(\theta -t)^{3}}}\right];t\in \mathbb {R} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5fe7cfd9def2afd3dfa51dda1623239c24a18f5)
![{\displaystyle \mu _{r}'={\frac {r![2\theta +(r+1)(r+2)]}{2\theta ^{r}(1+\theta )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a82aacde78c15ef02ad43daa1b8581eab2bd5104)
![{\displaystyle \operatorname {E} [X]={\frac {(\theta +3)}{\theta (1+\theta )}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7952b490b76d379ef0b54b4db39e1d46faedd917)
![{\displaystyle \mu _{r}=E[{(X-\mu )}^{r}]=\sum _{j=0}^{r}{\binom {r}{j}}\mu _{r}{'}(-{\mu })^{r-j},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a317c1a2538e203734aa6b1ee12837b1553fe87)
![{\displaystyle \operatorname {Var} [X]={\frac {(\theta ^{2}+8\theta +3)}{\theta ^{2}(1+\theta )^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3815f0355e86cb3aaf2c02a56ee3bd1022974b8)
![{\displaystyle Mode[X]={\begin{cases}{\frac {1+{\sqrt {1-2\theta }}}{\theta }}&{\text{if}}&0<\theta \leq 1/2,\\0&{\text{otherwise}}.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/657924df38cc96977bd1f23f032f2e2ee4ab5656)
![{\displaystyle \lim _{t\to 0}m(t;\theta )=E[X]={\frac {(\theta +3)}{\theta (1+\theta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd14e227115ed4bb5a6cd7166d177626426f580)
