Continuous probability distribution
The Kaniadakis Weibull distribution (or κ -Weibull distribution)probability distribution arising as a generalization of the Weibull distribution .[ 1] [ 2] Kaniadakis κ -distribution . The κ-Weibull distribution has been adopted successfully for describing a wide variety of complex systems in seismology, economy, epidemiology, among many others.
Definitions
Probability density function
The Kaniadakis κ -Weibull distribution is exhibits power-law right tails, and it has the following probability density function :[ 3]
f
κ κ -->
(
x
)
=
α α -->
β β -->
x
α α -->
− − -->
1
1
+
κ κ -->
2
β β -->
2
x
2
α α -->
exp
κ κ -->
-->
(
− − -->
β β -->
x
α α -->
)
{\displaystyle f_{_{\kappa }}(x)={\frac {\alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}\exp _{\kappa }(-\beta x^{\alpha })}
valid for
x
≥ ≥ -->
0
{\displaystyle x\geq 0}
|
κ κ -->
|
<
1
{\displaystyle |\kappa |<1}
Kaniadakis entropy ,
β β -->
>
0
{\displaystyle \beta >0}
α α -->
>
0
{\displaystyle \alpha >0}
Weibull modulus .
The Weibull distribution is recovered as
κ κ -->
→ → -->
0.
{\displaystyle \kappa \rightarrow 0.}
Cumulative distribution function
The cumulative distribution function of κ -Weibull distribution is given by
F
κ κ -->
(
x
)
=
1
− − -->
exp
κ κ -->
-->
(
− − -->
β β -->
x
α α -->
)
{\displaystyle F_{\kappa }(x)=1-\exp _{\kappa }(-\beta x^{\alpha })}
valid for
x
≥ ≥ -->
0
{\displaystyle x\geq 0}
Weibull distribution is recovered in the classical limit
κ κ -->
→ → -->
0
{\displaystyle \kappa \rightarrow 0}
Survival distribution and hazard functions
The survival distribution function of κ -Weibull distribution is given by
S
κ κ -->
(
x
)
=
exp
κ κ -->
-->
(
− − -->
β β -->
x
α α -->
)
{\displaystyle S_{\kappa }(x)=\exp _{\kappa }(-\beta x^{\alpha })}
valid for
x
≥ ≥ -->
0
{\displaystyle x\geq 0}
Weibull distribution is recovered in the classical limit
κ κ -->
→ → -->
0
{\displaystyle \kappa \rightarrow 0}
Comparison between the Kaniadakis κ-Weibull probability function and its cumulative.
The hazard function of the κ -Weibull distribution is obtained through the solution of the κ -rate equation:
S
κ κ -->
(
x
)
d
x
=
− − -->
h
κ κ -->
S
κ κ -->
(
x
)
{\displaystyle {\frac {S_{\kappa }(x)}{dx}}=-h_{\kappa }S_{\kappa }(x)}
with
S
κ κ -->
(
0
)
=
1
{\displaystyle S_{\kappa }(0)=1}
h
κ κ -->
{\displaystyle h_{\kappa }}
h
κ κ -->
=
α α -->
β β -->
x
α α -->
− − -->
1
1
+
κ κ -->
2
β β -->
2
x
2
α α -->
{\displaystyle h_{\kappa }={\frac {\alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}}
The cumulative κ -Weibull distribution is related to the κ -hazard function by the following expression:
S
κ κ -->
=
e
− − -->
H
κ κ -->
(
x
)
{\displaystyle S_{\kappa }=e^{-H_{\kappa }(x)}}
where
H
κ κ -->
(
x
)
=
∫ ∫ -->
0
x
h
κ κ -->
(
z
)
d
z
{\displaystyle H_{\kappa }(x)=\int _{0}^{x}h_{\kappa }(z)dz}
H
κ κ -->
(
x
)
=
1
κ κ -->
arcsinh
(
κ κ -->
β β -->
x
α α -->
)
{\displaystyle H_{\kappa }(x)={\frac {1}{\kappa }}{\textrm {arcsinh}}\left(\kappa \beta x^{\alpha }\right)}
is the cumulative κ -hazard function. The cumulative hazard function of the Weibull distribution is recovered in the classical limit
κ κ -->
→ → -->
0
{\displaystyle \kappa \rightarrow 0}
H
(
x
)
=
β β -->
x
α α -->
{\displaystyle H(x)=\beta x^{\alpha }}
Properties
The κ -Weibull distribution has moment of order
m
∈ ∈ -->
N
{\displaystyle m\in \mathbb {N} }
E
-->
[
X
m
]
=
|
2
κ κ -->
β β -->
|
− − -->
m
/
α α -->
1
+
κ κ -->
m
α α -->
Γ Γ -->
(
1
2
κ κ -->
− − -->
m
2
α α -->
)
Γ Γ -->
(
1
2
κ κ -->
+
m
2
α α -->
)
Γ Γ -->
(
1
+
m
α α -->
)
{\displaystyle \operatorname {E} [X^{m}]={\frac {|2\kappa \beta |^{-m/\alpha }}{1+\kappa {\frac {m}{\alpha }}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}\Gamma {\Big (}1+{\frac {m}{\alpha }}{\Big )}}
The median and the mode are:
x
median
(
F
κ κ -->
)
=
β β -->
− − -->
1
/
α α -->
(
ln
κ κ -->
-->
(
2
)
)
1
/
α α -->
{\displaystyle x_{\textrm {median}}(F_{\kappa })=\beta ^{-1/\alpha }{\Bigg (}\ln _{\kappa }(2){\Bigg )}^{1/\alpha }}
x
mode
=
β β -->
− − -->
1
/
α α -->
(
α α -->
2
+
2
κ κ -->
2
(
α α -->
− − -->
1
)
2
κ κ -->
2
(
α α -->
2
− − -->
κ κ -->
2
)
)
1
/
2
α α -->
(
1
+
4
κ κ -->
2
(
α α -->
2
− − -->
κ κ -->
2
)
(
α α -->
− − -->
1
)
2
[
α α -->
2
+
2
κ κ -->
2
(
α α -->
− − -->
1
)
]
2
− − -->
1
)
1
/
2
α α -->
(
α α -->
>
1
)
{\displaystyle x_{\textrm {mode}}=\beta ^{-1/\alpha }{\Bigg (}{\frac {\alpha ^{2}+2\kappa ^{2}(\alpha -1)}{2\kappa ^{2}(\alpha ^{2}-\kappa ^{2})}}{\Bigg )}^{1/2\alpha }{\Bigg (}{\sqrt {1+{\frac {4\kappa ^{2}(\alpha ^{2}-\kappa ^{2})(\alpha -1)^{2}}{[\alpha ^{2}+2\kappa ^{2}(\alpha -1)]^{2}}}}}-1{\Bigg )}^{1/2\alpha }\quad (\alpha >1)}
Quantiles
The quantiles are given by the following expression
x
quantile
(
F
κ κ -->
)
=
β β -->
− − -->
1
/
α α -->
[
ln
κ κ -->
-->
(
1
1
− − -->
F
κ κ -->
)
]
1
/
α α -->
{\displaystyle x_{\textrm {quantile}}(F_{\kappa })=\beta ^{-1/\alpha }{\Bigg [}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )}{\Bigg ]}^{1/\alpha }}
with
0
≤ ≤ -->
F
κ κ -->
≤ ≤ -->
1
{\displaystyle 0\leq F_{\kappa }\leq 1}
Gini coefficient
The Gini coefficient is:[ 3]
G
κ κ -->
=
1
− − -->
α α -->
+
κ κ -->
α α -->
+
1
2
κ κ -->
Γ Γ -->
(
1
κ κ -->
− − -->
1
2
α α -->
)
Γ Γ -->
(
1
κ κ -->
+
1
2
α α -->
)
Γ Γ -->
(
1
2
κ κ -->
+
1
2
α α -->
)
Γ Γ -->
(
1
2
κ κ -->
− − -->
1
2
α α -->
)
{\displaystyle \operatorname {G} _{\kappa }=1-{\frac {\alpha +\kappa }{\alpha +{\frac {1}{2}}\kappa }}{\frac {\Gamma {\Big (}{\frac {1}{\kappa }}-{\frac {1}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {1}{2\alpha }}{\Big )}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{2\alpha }}{\Big )}}}}
Asymptotic behavior
The κ -Weibull distribution II behaves asymptotically as follows:[ 3]
lim
x
→ → -->
+
∞ ∞ -->
f
κ κ -->
(
x
)
∼ ∼ -->
α α -->
κ κ -->
(
2
κ κ -->
β β -->
)
− − -->
1
/
κ κ -->
x
− − -->
1
− − -->
α α -->
/
κ κ -->
{\displaystyle \lim _{x\to +\infty }f_{\kappa }(x)\sim {\frac {\alpha }{\kappa }}(2\kappa \beta )^{-1/\kappa }x^{-1-\alpha /\kappa }}
lim
x
→ → -->
0
+
f
κ κ -->
(
x
)
=
α α -->
β β -->
x
α α -->
− − -->
1
{\displaystyle \lim _{x\to 0^{+}}f_{\kappa }(x)=\alpha \beta x^{\alpha -1}}
The κ -Weibull distribution is a generalization of:
A κ -Weibull distribution corresponds to a κ -deformed Rayleigh distribution when
α α -->
=
2
{\displaystyle \alpha =2}
Rayleigh distribution when
κ κ -->
=
0
{\displaystyle \kappa =0}
α α -->
=
2
{\displaystyle \alpha =2}
Applications
The κ -Weibull distribution has been applied in several areas, such as:
In economy , for analyzing personal income models , in order to accurately describing simultaneously the income distribution among the richest part and the great majority of the population.[ 1] [ 4] [ 5]
In seismology , the κ-Weibull represents the statistical distribution of magnitude of the earthquakes distributed across the Earth, generalizing the Gutenberg–Richter law ,[ 6] [ 7] [ 8]
In epidemiology , the κ-Weibull distribution presents a universal feature for epidemiological analysis.[ 9]
See also
References
^ a b Clementi, F.; Gallegati, M.; Kaniadakis, G. (2007). "κ-generalized statistics in personal income distribution" . The European Physical Journal B . 57 (2): 187– 193. arXiv :physics/0607293 Bibcode :2007EPJB...57..187C . doi :10.1140/epjb/e2007-00120-9 . ISSN 1434-6028 . S2CID 15777288 . ^ Clementi, F.; Di Matteo, T. ; Gallegati, M.; Kaniadakis, G. (2008). "The -generalized distribution: A new descriptive model for the size distribution of incomes" . Physica A: Statistical Mechanics and Its Applications . 387 (13): 3201– 3208. arXiv :0710.3645 doi :10.1016/j.physa.2008.01.109 . S2CID 2590064 . ^ 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 . ^ Clementi, Fabio; Gallegati, Mauro; Kaniadakis, Giorgio (October 2010). "A model of personal income distribution with application to Italian data" . Empirical Economics . 39 (2): 559– 591. doi :10.1007/s00181-009-0318-2 . ISSN 0377-7332 . S2CID 154273794 . ^ Clementi, F; Gallegati, M; Kaniadakis, G (2012-12-06). "A generalized statistical model for the size distribution of wealth" . Journal of Statistical Mechanics: Theory and Experiment . 2012 (12): P12006. arXiv :1209.4787 Bibcode :2012JSMTE..12..006C . doi :10.1088/1742-5468/2012/12/P12006 . ISSN 1742-5468 . S2CID 18961951 . ^ da Silva, Sérgio Luiz E.F. (2021). "κ -generalised Gutenberg–Richter law and the self-similarity of earthquakes" . Chaos, Solitons & Fractals . 143 : 110622. Bibcode :2021CSF...14310622D . doi :10.1016/j.chaos.2020.110622 . S2CID 234063959 . ^ Hristopulos, Dionissios T.; Petrakis, Manolis P.; Kaniadakis, Giorgio (2014-05-28). "Finite-size effects on return interval distributions for weakest-link-scaling systems" . Physical Review E . 89 (5): 052142. arXiv :1308.1881 Bibcode :2014PhRvE..89e2142H . doi :10.1103/PhysRevE.89.052142 . ISSN 1539-3755 . PMID 25353774 . S2CID 22310350 . ^ Hristopulos, Dionissios; Petrakis, Manolis; Kaniadakis, Giorgio (2015-03-09). "Weakest-Link Scaling and Extreme Events in Finite-Sized Systems" . Entropy . 17 (3): 1103– 1122. Bibcode :2015Entrp..17.1103H . doi :10.3390/e17031103 ISSN 1099-4300 . ^ Kaniadakis, Giorgio; Baldi, Mauro M.; Deisboeck, Thomas S.; Grisolia, Giulia; Hristopulos, Dionissios T.; Scarfone, Antonio M.; Sparavigna, Amelia; Wada, Tatsuaki; Lucia, Umberto (2020). "The κ-statistics approach to epidemiology" . Scientific Reports . 10 (1): 19949. Bibcode :2020NatSR..1019949K . doi :10.1038/s41598-020-76673-3 . ISSN 2045-2322 . PMC 7673996 PMID 33203913 .
External links
![{\displaystyle \beta ^{-1/\alpha }{\Bigg [}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )}{\Bigg ]}^{1/\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b52bba751bd8c84676ad4ea48dde2c717b197e4)
![{\displaystyle \beta ^{-1/\alpha }{\Bigg (}{\frac {\alpha ^{2}+2\kappa ^{2}(\alpha -1)}{2\kappa ^{2}(\alpha ^{2}-\kappa ^{2})}}{\sqrt {1+{\frac {4\kappa ^{2}(\alpha ^{2}-\kappa ^{2})(\alpha -1)^{2}}{[\alpha ^{2}+2\kappa ^{2}(\alpha -1)]^{2}}}}}-1{\Bigg )}^{1/2\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e553db64cc7631a2e8bd64f6d31b670a13ffb52b)
![{\displaystyle \operatorname {E} [X^{m}]={\frac {|2\kappa \beta |^{-m/\alpha }}{1+\kappa {\frac {m}{\alpha }}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}\Gamma {\Big (}1+{\frac {m}{\alpha }}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a910cb084f83179aebeac519e1ae0468889a2f)
![{\displaystyle x_{\textrm {mode}}=\beta ^{-1/\alpha }{\Bigg (}{\frac {\alpha ^{2}+2\kappa ^{2}(\alpha -1)}{2\kappa ^{2}(\alpha ^{2}-\kappa ^{2})}}{\Bigg )}^{1/2\alpha }{\Bigg (}{\sqrt {1+{\frac {4\kappa ^{2}(\alpha ^{2}-\kappa ^{2})(\alpha -1)^{2}}{[\alpha ^{2}+2\kappa ^{2}(\alpha -1)]^{2}}}}}-1{\Bigg )}^{1/2\alpha }\quad (\alpha >1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72be2202fcd748ab4d08b08385521ed8ce37d1b0)
![{\displaystyle x_{\textrm {quantile}}(F_{\kappa })=\beta ^{-1/\alpha }{\Bigg [}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )}{\Bigg ]}^{1/\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/142dbdd25f9a709d2e3bfb7bd64030643787bfa8)
