From Wikipedia, the free encyclopedia
Aitchinson distribution Notation
A
i
t
c
h
i
n
s
o
n
(
a
,
b
,
θ
)
{\displaystyle \mathrm {Aitchinson} \left(a,b,\theta \right)}
Support
x ∈ { 0, 1, 2 , ... } PMF
{\displaystyle }
CDF
{\displaystyle }
Mean
{\displaystyle }
Median
{\displaystyle }
Mode
{\displaystyle }
Variance
{\displaystyle }
Skewness
{\displaystyle }
Excess kurtosis
{\displaystyle }
Entropy
{\displaystyle }
MGF
{\displaystyle }
CF
{\displaystyle }
PGF
{\displaystyle }
probability mass function of the Aitchinson distribution is given by
A
i
t
c
h
i
n
s
o
n
(
x
;
a
,
b
,
θ
)
≡
Pr
(
X
=
x
)
=
e
−
a
∑
j
=
0
−
2
k
+
x
∑
k
=
0
[
x
2
]
(
b
x
)
−
j
−
2
k
+
x
(
−
a
e
−
b
(
−
1
+
θ
)
)
k
(
a
θ
)
j
j
!
k
!
(
−
j
−
2
k
+
x
)
!
{\displaystyle \mathrm {Aitchinson} \left(x;a,b,\theta \right)\equiv \Pr(X=x)=e^{-a}\sum _{j=0}^{-2k+x}\sum _{k=0}^{\left[{\frac {x}{2}}\right]}{\frac {(bx)^{-j-2k+x}\left(-ae^{-b}(-1+\theta )\right)^{k}(a\theta )^{j}}{j!k!(-j-2k+x)!}}}
x
=
0
,
1
,
2
,
…
{\displaystyle x=0,1,2,\dots }
a
,
b
≥
0
{\displaystyle a,b\geq 0}
0
≤
θ
≤
1
{\displaystyle 0\leq \theta \leq 1}
[
z
]
=
integer part of z
{\displaystyle \left[z\right]={\text{integer part of z}}}
Expected Value
E
[
X
]
=
−
a
(
b
(
θ
−
1
)
+
θ
−
2
)
{\displaystyle \operatorname {E} [X]=-a(b(\theta -1)+\theta -2)}
Variance
Var
(
X
)
=
a
(
−
b
(
b
+
5
)
(
θ
−
1
)
−
3
θ
+
4
)
{\displaystyle \operatorname {Var} (X)=a(-b(b+5)(\theta -1)-3\theta +4)}
Moment Generating Function
M
X
(
t
)
=
exp
(
−
a
(
(
θ
−
1
)
e
b
(
e
t
−
1
)
+
2
t
−
θ
e
t
+
1
)
)
{\displaystyle M_{X}(t)=\exp \left(-a\left((\theta -1)e^{b\left(e^{t}-1\right)+2t}-\theta e^{t}+1\right)\right)}
Characteristic Function
φ
X
(
t
)
=
exp
(
−
a
(
(
θ
−
1
)
e
b
(
e
i
t
−
1
)
+
2
i
t
−
θ
e
i
t
+
1
)
)
{\displaystyle \varphi _{X}(t)=\exp \left(-a\left((\theta -1)e^{b\left(e^{it}-1\right)+2it}-\theta e^{it}+1\right)\right)}
Probability Generating Function
G
(
t
)
=
exp
(
−
a
(
(
θ
−
1
)
t
2
e
b
(
t
−
1
)
−
θ
t
+
1
)
)
{\displaystyle G(t)=\exp \left(-a\left((\theta -1)t^{2}e^{b(t-1)}-\theta t+1\right)\right)}
Interrelations [ edit ]
Symbol
Meaning
∼
{\displaystyle \sim }
X
∼
Y
{\displaystyle X\sim Y}
≡
{\displaystyle \equiv }
the distribution in the title is identical with this distribution
⇐
{\displaystyle \Leftarrow }
the distribution in title is a special case of this distribution
⇒
{\displaystyle \Rightarrow }
this distribution is a special case of the distribution in the title
←
{\displaystyle \leftarrow }
this distribution converges to the distribution in the title
→
{\displaystyle \rightarrow }
the distribution in the title converges to this distribution
Relationship
Distribution
When
≡
{\displaystyle \equiv }
Poisson
(
a
)
{\displaystyle (a)}
∨
{\displaystyle \vee }
(
b
,
θ
)
{\displaystyle \left(b,\theta \right)}
P
x
=
{
θ
x
=
1
e
−
b
(
1
−
θ
)
b
x
−
2
(
x
−
2
)
!
x
=
2
,
3
,
…
b
≥
0
0
≤
θ
≤
1
{\displaystyle P_{x}={\begin{cases}\theta &x=1\\{\frac {e^{-b}(1-\theta )b^{x-2}}{(x-2)!}}&x=2,3,\dots \\\end{cases}}\qquad b\geq 0\qquad 0\leq \theta \leq 1}
⇐
{\displaystyle \Leftarrow }
generalized Poisson family
(
a
,
H
(
.
)
)
{\displaystyle \left(a,H(.)\right)}
←
{\displaystyle \leftarrow }
multiple Poisson
(
n
,
a
i
)
{\displaystyle \left(n,a_{i}\right)}
n
→
∞
a
i
=
{
a
θ
i
=
1
a
b
−
2
+
i
e
−
b
(
1
−
θ
)
(
−
2
+
i
)
!
i
=
2
,
3
,
…
{\displaystyle n\rightarrow \infty \qquad a_{i}={\begin{cases}a\theta &i=1\\{\frac {ab^{-2+i}e^{-b}(1-\theta )}{(-2+i)!}}&i=2,3,\dots \\\end{cases}}}
⇒
{\displaystyle \Rightarrow }
deterministic
(
0
)
{\displaystyle \left(0\right)}
a
=
0
{\displaystyle a=0}
⇒
{\displaystyle \Rightarrow }
Hermite
(
a
′
,
b
′
)
{\displaystyle \left(a',b'\right)}
a
=
a
′
+
b
′
θ
=
a
′
a
′
+
b
′
b
=
0
{\displaystyle a=a'+b'\qquad \theta ={\frac {a'}{a'+b'}}\qquad b=0}
⇒
{\displaystyle \Rightarrow }
Hirata-Poisson
(
a
′
,
b
′
)
{\displaystyle \left(a',b'\right)}
θ
=
1
−
b
′
b
=
0
{\displaystyle \theta =1-b'\qquad b=0}
⇒
{\displaystyle \Rightarrow }
Poisson
(
a
)
{\displaystyle \left(a\right)}
θ
=
1
{\displaystyle \theta =1}
References [ edit ] Aitchinson, J. (1955). On the distribution of a positive random variable having a distribution probability mass at the origin J. of the American Statistical Association 50, 901-908 Kupper, J. (1960-62). Wahrscheinlich-keitstheoretische Modelle in der Schadenversicherung. Teil I: Die Schadenzahl. Blätter der Deutschen Gesellschaft für Versicherungsmathematik 5, 451-503. Wimmer, G., Altmann. (1996a). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical J. 8, 995-1011. Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999) , pg 7
![{\displaystyle \mathrm {Aitchinson} \left(x;a,b,\theta \right)\equiv \Pr(X=x)=e^{-a}\sum _{j=0}^{-2k+x}\sum _{k=0}^{\left[{\frac {x}{2}}\right]}{\frac {(bx)^{-j-2k+x}\left(-ae^{-b}(-1+\theta )\right)^{k}(a\theta )^{j}}{j!k!(-j-2k+x)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1551b83b606c25c4b6d3ebc10a7e5c2e463a7f)
![{\displaystyle \left[z\right]={\text{integer part of z}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7556d0532707b8ddc2bd018942024c99c0f06413)
![{\displaystyle \operatorname {E} [X]=-a(b(\theta -1)+\theta -2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edae3c6a830b778bd3670528e49bcd8f4c769f53)
