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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
( R e d i r e c t e d f r o m K a n i a d a k i s L o g i s t i c d i s t r i b u t i o n )
The Kaniadakis Logistic distribution (also known as κ- Logisticdistribution) is a generalized version of the Logistic distribution associated with the Kaniadakis statistics . It is one example of a Kaniadakis distribution . The κ-Logistic probability distribution describes the population kinetics behavior of bosonic (
0
<
λ
<
1
{\displaystyle 0<\lambda <1}
) or fermionic (
λ
>
1
{\displaystyle \lambda >1}
) character.[1]
Definitions
[ edit ]
Probability density function
[ edit ]
The Kaniadakis κ -Logistic distribution is a four-parameter family of continuous statistical distributions , which is part of a class of statistical distributions emerging from the Kaniadakis κ-statistics . This distribution has the following probability density function :[1]
f
κ
(
x
)
=
λ
α
β
x
α
−
1
1
+
κ
2
β
2
x
2
α
exp
κ
(
−
β
x
α
)
[
1
+
(
λ
−
1
)
exp
κ
(
−
β
x
α
)
]
2
{\displaystyle f_{_{\kappa }}(x )={\frac {\lambda \alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}{\frac {\exp _{\kappa }(-\beta x^{\alpha })}{[1+(\lambda -1)\exp _{\kappa }(-\beta x^{\alpha })]^{2}}}}
valid for
x
≥
0
{\displaystyle x\geq 0}
, where
0
≤
|
κ
|
<
1
{\displaystyle 0\leq |\kappa |<1}
is the entropic index associated with the Kaniadakis entropy ,
β
>
0
{\displaystyle \beta >0}
is the rate parameter ,
λ
>
0
{\displaystyle \lambda >0}
, and
α
>
0
{\displaystyle \alpha >0}
is the shape parameter.
The Logistic distribution is recovered as
κ
→
0.
{\displaystyle \kappa \rightarrow 0.}
Cumulative distribution function
[ edit ]
The cumulative distribution function of κ -Logistic is given by
F
κ
(
x
)
=
1
−
exp
κ
(
−
β
x
α
)
1
+
(
λ
−
1
)
exp
κ
(
−
β
x
α
)
{\displaystyle F_{\kappa }(x )={\frac {1-\exp _{\kappa }(-\beta x^{\alpha })}{1+(\lambda -1)\exp _{\kappa }(-\beta x^{\alpha })}}}
valid for
x
≥
0
{\displaystyle x\geq 0}
. The cumulative Logistic distribution is recovered in the classical limit
κ
→
0
{\displaystyle \kappa \rightarrow 0}
.
Survival and hazard functions
[ edit ]
The survival distribution function of κ -Logistic distribution is given by
S
κ
(
x
)
=
λ
exp
κ
(
β
x
α
)
+
λ
−
1
{\displaystyle S_{\kappa }(x )={\frac {\lambda }{\exp _{\kappa }(\beta x^{\alpha })+\lambda -1}}}
valid for
x
≥
0
{\displaystyle x\geq 0}
. The survival Logistic distribution is recovered in the classical limit
κ
→
0
{\displaystyle \kappa \rightarrow 0}
.
The hazard function associated with the κ -Logistic distribution is obtained by the solution of the following evolution equation:
S
κ
(
x
)
d
x
=
−
h
κ
S
κ
(
x
)
(
1
−
λ
−
1
λ
S
κ
(
x
)
)
{\displaystyle {\frac {S_{\kappa }(x )}{dx}}=-h_{\kappa }S_{\kappa }(x )\left(1-{\frac {\lambda -1}{\lambda }}S_{\kappa }(x )\right)}
with
S
κ
(
0
)
=
1
{\displaystyle S_{\kappa }(0)=1}
, where
h
κ
{\displaystyle h_{\kappa }}
is the hazard function:
h
κ
=
α
β
x
α
−
1
1
+
κ
2
β
2
x
2
α
{\displaystyle h_{\kappa }={\frac {\alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}}
The cumulative Kaniadakis κ -Logistic distribution is related to the hazard function by the following expression:
S
κ
=
e
−
H
κ
(
x
)
{\displaystyle S_{\kappa }=e^{-H_{\kappa }(x )}}
where
H
κ
(
x
)
=
∫
0
x
h
κ
(
z
)
d
z
{\displaystyle H_{\kappa }(x )=\int _{0}^{x}h_{\kappa }(z )dz}
is the cumulative hazard function. The cumulative hazard function of the Logistic distribution is recovered in the classical limit
κ
→
0
{\displaystyle \kappa \rightarrow 0}
.
[ edit ]
The survival function of the κ -Logistic distribution represents the κ -deformation of the Fermi-Dirac function, and becomes a Fermi-Dirac distribution in the classical limit
κ
→
0
{\displaystyle \kappa \rightarrow 0}
.[1]
The κ -Logistic distribution is a generalization of the κ -Weibull distribution when
λ
=
1
{\displaystyle \lambda =1}
.
A κ -Logistic distribution corresponds to a Half-Logistic distribution when
λ
=
2
{\displaystyle \lambda =2}
,
α
=
1
{\displaystyle \alpha =1}
and
κ
=
0
{\displaystyle \kappa =0}
.
The ordinary Logistic distribution is a particular case of a κ -Logistic distribution, when
κ
=
0
{\displaystyle \kappa =0}
.
Applications
[ edit ]
The κ -Logistic distribution has been applied in several areas, such as:
See also
[ edit ]
References
[ edit ]
^ Kaniadakis, G. (2001). "H-theorem and generalized entropies within the framework of nonlinear kinetics" . Physics Letters A . 288 (5–6): 283–291. arXiv :cond-mat/0109192 . Bibcode :2001PhLA..288..283K . doi :10.1016/S0375-9601(01 )00543-6 . S2CID 119445915 .
^ Lourek, Imene; Tribeche, Mouloud (2017). "Thermodynamic properties of the blackbody radiation: A Kaniadakis approach" . Physics Letters A . 381 (5 ): 452–456. Bibcode :2017PhLA..381..452L . doi :10.1016/j.physleta.2016.12.019 .
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_logistic_distribution&oldid=1155477071 "
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