Continuous probability distribution
The Kaniadakis Gaussian distribution (also known as κ -Gaussian distribution) is a
probability distribution which arises as a generalization of the
Gaussian distribution from the maximization of the Kaniadakis entropy under appropriated constraints. It is one example of a Kaniadakis κ -distribution. The κ-Gaussian distribution has been applied successfully for describing several complex systems in economy,
[1] geophysics,
[2] astrophysics, among many others.
The κ-Gaussian distribution is a particular case of the
κ-Generalized Gamma distribution .
[3]
Definitions
Probability density function
The general form of the centered Kaniadakis κ -Gaussian probability density function is:
[3]
f
κ
(
x
)
=
Z
κ
exp
κ
(
−
β
x
2
)
{\displaystyle f_{_{\kappa }}(x)=Z_{\kappa }\exp _{\kappa }(-\beta x^{2})}
where
|
κ
|
<
1
{\displaystyle |\kappa |<1}
is the entropic index associated with the
Kaniadakis entropy ,
β
>
0
{\displaystyle \beta >0}
is the scale parameter, and
Z
κ
=
2
β
κ
π
(
1
+
1
2
κ
)
Γ
(
1
2
κ
+
1
4
)
Γ
(
1
2
κ
−
1
4
)
{\displaystyle Z_{\kappa }={\sqrt {\frac {2\beta \kappa }{\pi }}}{\Bigg (}1+{\frac {1}{2}}\kappa {\Bigg )}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}}
is the normalization constant.
The
standard Normal distribution is recovered in the limit
κ
→
0.
{\displaystyle \kappa \rightarrow 0.}
Cumulative distribution function
The
cumulative distribution function of κ -Gaussian distribution is given by
F
κ
(
x
)
=
1
2
+
1
2
erf
κ
(
β
x
)
{\displaystyle F_{\kappa }(x)={\frac {1}{2}}+{\frac {1}{2}}{\textrm {erf}}_{\kappa }{\big (}{\sqrt {\beta }}x{\big )}}
where
erf
κ
(
x
)
=
(
2
+
κ
)
2
κ
π
Γ
(
1
2
κ
+
1
4
)
Γ
(
1
2
κ
−
1
4
)
∫
0
x
exp
κ
(
−
t
2
)
d
t
{\displaystyle {\textrm {erf}}_{\kappa }(x)={\Big (}2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi }}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt}
is the Kaniadakis κ -Error function, which is a generalization of the ordinary
Error function
erf
(
x
)
{\displaystyle {\textrm {erf}}(x)}
as
κ
→
0
{\displaystyle \kappa \rightarrow 0}
.
Properties
Moments, mean and variance
The centered κ -Gaussian distribution has a moment of odd order equal to zero, including the mean.
The variance is finite for
κ
<
2
/
3
{\displaystyle \kappa <2/3}
and is given by:
Var
X
=
σ
κ
2
=
1
β
2
+
κ
2
−
κ
4
κ
4
−
9
κ
2
Γ
(
1
2
κ
+
1
4
)
Γ
(
1
2
κ
−
1
4
)
2
{\displaystyle \operatorname {Var} [X]=\sigma _{\kappa }^{2}={\frac {1}{\beta }}{\frac {2+\kappa }{2-\kappa }}{\frac {4\kappa }{4-9\kappa ^{2}}}\left[{\frac {\Gamma \left({\frac {1}{2\kappa }}+{\frac {1}{4}}\right)}{\Gamma \left({\frac {1}{2\kappa }}-{\frac {1}{4}}\right)}}\right]^{2}}
Kurtosis
The
kurtosis of the centered κ -Gaussian distribution may be computed thought:
Kurt
X
=
E
X
4
σ
κ
4
{\displaystyle \operatorname {Kurt} [X]=\operatorname {E} \left[{\frac {X^{4}}{\sigma _{\kappa }^{4}}}\right]}
which can be written as
Kurt
X
=
2
Z
κ
σ
κ
4
∫
0
∞
x
4
exp
κ
(
−
β
x
2
)
d
x
{\displaystyle \operatorname {Kurt} [X]={\frac {2Z_{\kappa }}{\sigma _{\kappa }^{4}}}\int _{0}^{\infty }x^{4}\,\exp _{\kappa }\left(-\beta x^{2}\right)dx}
Thus, the
kurtosis of the centered κ -Gaussian distribution is given by:
Kurt
X
=
3
π
Z
κ
2
β
2
/
3
σ
κ
4
|
2
κ
|
−
5
/
2
1
+
5
2
|
κ
|
Γ
(
1
|
2
κ
|
−
5
4
)
Γ
(
1
|
2
κ
|
+
5
4
)
{\displaystyle \operatorname {Kurt} [X]={\frac {3{\sqrt {\pi }}Z_{\kappa }}{2\beta ^{2/3}\sigma _{\kappa }^{4}}}{\frac {|2\kappa |^{-5/2}}{1+{\frac {5}{2}}|\kappa |}}{\frac {\Gamma \left({\frac {1}{|2\kappa |}}-{\frac {5}{4}}\right)}{\Gamma \left({\frac {1}{|2\kappa |}}+{\frac {5}{4}}\right)}}}
or
Kurt
X
=
3
β
11
/
6
2
κ
2
|
2
κ
|
−
5
/
2
1
+
5
2
|
κ
|
(
1
+
1
2
κ
)
(
2
−
κ
2
+
κ
)
2
(
4
−
9
κ
2
4
κ
)
2
Γ
(
1
2
κ
−
1
4
)
Γ
(
1
2
κ
+
1
4
)
3
Γ
(
1
|
2
κ
|
−
5
4
)
Γ
(
1
|
2
κ
|
+
5
4
)
{\displaystyle \operatorname {Kurt} [X]={\frac {3\beta ^{11/6}{\sqrt {2\kappa }}}{2}}{\frac {|2\kappa |^{-5/2}}{1+{\frac {5}{2}}|\kappa |}}{\Bigg (}1+{\frac {1}{2}}\kappa {\Bigg )}\left({\frac {2-\kappa }{2+\kappa }}\right)^{2}\left({\frac {4-9\kappa ^{2}}{4\kappa }}\right)^{2}\left[{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}}\right]^{3}{\frac {\Gamma \left({\frac {1}{|2\kappa |}}-{\frac {5}{4}}\right)}{\Gamma \left({\frac {1}{|2\kappa |}}+{\frac {5}{4}}\right)}}}
κ-Error function
The Kaniadakis κ -Error function (or κ -Error function ) is a one-parameter generalization of the
ordinary error function defined as:
[3]
erf
κ
(
x
)
=
(
2
+
κ
)
2
κ
π
Γ
(
1
2
κ
+
1
4
)
Γ
(
1
2
κ
−
1
4
)
∫
0
x
exp
κ
(
−
t
2
)
d
t
{\displaystyle \operatorname {erf} _{\kappa }(x)={\Big (}2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi }}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt}
Although the error function cannot be expressed in terms of elementary functions, numerical approximations are commonly employed.
For a
random variable X distributed according to a κ-Gaussian distribution with mean 0 and standard deviation
β
{\displaystyle {\sqrt {\beta }}}
, κ-Error function means the probability that X falls in the interval
−
x
,
x
{\displaystyle [-x,\,x]}
.
Applications
The κ -Gaussian distribution has been applied in several areas, such as:
See also
References
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doi :
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ISSN
1434-6028 .
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254116243 .
^
a
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doi :
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^
a
b
c Kaniadakis, G. (2021-01-01).
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arXiv :
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