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( T o p )
1
D e f i n i t i o n s
T o g g l e D e f i n i t i o n s s u b s e c t i o n
1 . 1
P r o b a b i l i t y d e n s i t y f u n c t i o n
1 . 2
C u m u l a t i v e d i s t r i b u t i o n f u n c t i o n
2
P r o p e r t i e s
T o g g l e P r o p e r t i e s s u b s e c t i o n
2 . 1
M o m e n t s , m e a n a n d v a r i a n c e
2 . 2
K u r t o s i s
3
κ - E r r o r f u n c t i o n
4
A p p l i c a t i o n s
5
S e e a l s o
6
R e f e r e n c e s
7
E x t e r n a l l i n k s
T o g g l e t h e t a b l e o f c o n t e n t s
K a n i a d a k i s G a u s s i a n d i s t r i b u t i o n
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P r i n t / e x p o r t
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
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
[ edit ]
Probability density function
[ edit ]
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
[ edit ]
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
[ edit ]
Moments, mean and variance
[ edit ]
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
[ edit ]
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
[ edit ]
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
[ edit ]
The κ -Gaussian distribution has been applied in several areas, such as:
See also
[ edit ]
References
[ edit ]
^ a b c Kaniadakis, G. (2021-01-01). "New power-law tailed distributions emerging in κ-statistics (a )" . Europhysics Letters . 133 (1 ): 10002. arXiv :2203.01743 . Bibcode :2021EL....13310002K . doi :10.1209/0295-5075/133/10002 . ISSN 0295-5075 . S2CID 234144356 .
^ Moretto, Enrico; Pasquali, Sara; Trivellato, Barbara (2017). "A non-Gaussian option pricing model based on Kaniadakis exponential deformation" . The European Physical Journal B . 90 (10 ): 179. Bibcode :2017EPJB...90..179M . doi :10.1140/epjb/e2017-80112-x . ISSN 1434-6028 . S2CID 254116243 .
^ Wada, Tatsuaki; Suyari, Hiroki (2006). "κ-generalization of Gauss' law of error" . Physics Letters A . 348 (3–6): 89–93. arXiv :cond-mat/0505313 . Bibcode :2006PhLA..348...89W . doi :10.1016/j.physleta.2005.08.086 . S2CID 119003351 .
^ da Silva, Sérgio Luiz E.F.; Silva, R.; dos Santos Lima, Gustavo Z.; de Araújo, João M.; Corso, Gilberto (2022). "An outlier-resistant κ -generalized approach for robust physical parameter estimation" . Physica A: Statistical Mechanics and Its Applications . 600 : 127554. arXiv :2111.09921 . Bibcode :2022PhyA..60027554D . doi :10.1016/j.physa.2022.127554 . S2CID 248803855 .
^ Carvalho, J. C.; Silva, R.; do Nascimento jr., J. D.; Soares, B. B.; De Medeiros, J. R. (2010-09-01). "Observational measurement of open stellar clusters: A test of Kaniadakis and Tsallis statistics" . EPL (Europhysics Letters) . 91 (6 ): 69002. Bibcode :2010EL.....9169002C . doi :10.1209/0295-5075/91/69002 . ISSN 0295-5075 . S2CID 120902898 .
^ Carvalho, J. C.; Silva, R.; do Nascimento jr., J. D.; De Medeiros, J. R. (2008). "Power law statistics and stellar rotational velocities in the Pleiades" . EPL (Europhysics Letters) . 84 (5 ): 59001. arXiv :0903.0836 . Bibcode :2008EL.....8459001C . doi :10.1209/0295-5075/84/59001 . ISSN 0295-5075 . S2CID 7123391 .
^ Guedes, Guilherme; Gonçalves, Alessandro C.; Palma, Daniel A.P. (2017). "The Doppler Broadening Function using the Kaniadakis distribution" . Annals of Nuclear Energy . 110 : 453–458. doi :10.1016/j.anucene.2017.06.057 .
^ de Abreu, Willian V.; Gonçalves, Alessandro C.; Martinez, Aquilino S. (2019). "Analytical solution for the Doppler broadening function using the Kaniadakis distribution" . Annals of Nuclear Energy . 126 : 262–268. doi :10.1016/j.anucene.2018.11.023 . S2CID 125724227 .
^ Gougam, Leila Ait; Tribeche, Mouloud (2016). "Electron-acoustic waves in a plasma with a κ -deformed Kaniadakis electron distribution" . Physics of Plasmas . 23 (1 ): 014501. Bibcode :2016PhPl...23a4501G . doi :10.1063/1.4939477 . ISSN 1070-664X .
^ Chen, H.; Zhang, S. X.; Liu, S. Q. (2017). "The longitudinal plasmas modes of κ -deformed Kaniadakis distributed plasmas" . Physics of Plasmas . 24 (2 ): 022125. Bibcode :2017PhPl...24b2125C . doi :10.1063/1.4976992 . ISSN 1070-664X .
External links
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Kaniadakis_Gaussian_distribution&oldid=1155474799 "
C a t e g o r i e s :
● P r o b a b i l i t y d i s t r i b u t i o n s
● M a t h e m a t i c a l a n d q u a n t i t a t i v e m e t h o d s ( e c o n o m i c s )
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h t o p i c s o f u n c l e a r n o t a b i l i t y f r o m F e b r u a r y 2 0 2 3
● A l l a r t i c l e s w i t h t o p i c s o f u n c l e a r n o t a b i l i t y
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a
● T h i s p a g e w a s l a s t e d i t e d o n 1 8 M a y 2 0 2 3 , a t 0 9 : 3 4 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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