Kaniadakis Weibull distribution explained

κ-Weibull distribution
Type:	density
Parameters:	0 < \kappa < 1 \alpha > 0 rate shape (real) \beta> 0 rate (real)
Support:	x \in [ 0 , +inf ty) | pdf = (\alpha \beta x)/(\sqrt{ 1+ \kappa ^{2} \beta ^{2} x ^{2}) } \exp _{\kappa} ( - \beta x ^{\alpha} ) | cdf = 1 - \exp _{\kappa(-\beta} x ^{\alpha)} |pdf_image=File:Kaniadakis weibull pdf.png|cdf_image=File:Kaniadakis weibull cdf.png|moments=(( 2 \kappa \beta ) ^{-m/})/(1 + \kappa (m)/(\alpha)) (\Gamma ( (1)/(2 \kappa) - (m)/(2 \alpha) ))/(\Gamma ( (1)/(2 \kappa) + (m)/(2 \alpha) )) \Gamma ( 1+ (m)/(\alpha) ) |mode=\beta ( (\alpha ^{2} +2 \kappa ^{2} (\alpha -1 ))/(2 \kappa ^{2} ( \alpha ^{2} - \kappa ^{2)}) \sqrt{ 1 + (4 \kappa ^{2} (\alpha ^{2} - \kappa ^{2} )( \alpha -1 ) ^{2})/([ \alpha ^{2} +2 \kappa ^{2} (\alpha -1 ) ] ^{2}) } - 1 ) ^{1/2} |median=\beta ^{-1/\alpha} ( ln _{\kappa} ( 2 ) ) ^{1/\alpha} |quantile=\beta ^{-1} [ ln _{\kappa} ( (1)/(1- F _{\kappa}) ) ] ^{1/}

The Kaniadakis Weibull distribution (or κ-Weibull distribution) is a probability distribution arising as a generalization of the Weibull distribution.^{[1] [2]} It is one example of a 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 _{_{\kappa}} (x) = (\alpha \beta x ^{\alpha-1})/(\sqrt{ 1+ \kappa ^{2} \beta ^{2} x ^{2\alpha}) } \exp_\kappa(-\beta x^\alpha)

valid for

x \geq 0 , where |\kappa| < 1 is the entropic index associated with the Kaniadakis entropy, \beta > 0 is the scale parameter, and \alpha > 0 is the shape parameter or Weibull modulus.

The Weibull distribution is recovered as

\kappa → 0 .

Cumulative distribution function

The cumulative distribution function of κ-Weibull distribution is given by

F _{\kappa(x)} = 1 - \exp _{\kappa(-\beta} x ^{\alpha)}

valid for x \geq 0 . The cumulative Weibull distribution is recovered in the classical limit \kappa → 0 .

Survival distribution and hazard functions

The survival distribution function of κ-Weibull distribution is given by

S _{\kappa(x)} = \exp _{\kappa(-\beta} x ^{\alpha)}

valid for

x \geq 0 . The survival Weibull distribution is recovered in the classical limit \kappa → 0 .

The hazard function of the κ-Weibull distribution is obtained through the solution of the κ-rate equation:

(S _{\kappa(x)})/(dx) = - h _{\kappa} S _{\kappa(x)}

with S _{\kappa(0)} = 1 , where h _{\kappa} is the hazard function:h _{\kappa} = (\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 _{\kappa} = e ^{- H _{\kappa(x)}}

where

H _{\kappa} (x) = \int ^{x} _{0} h _{\kappa(z)} dz H _{\kappa} (x) = (1)/(\kappa) rm{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

\kappa → 0 : H(x) = \beta x ^{\alpha} .

Properties

Moments, median and mode

The κ-Weibull distribution has moment of order

m \in N given by\operatorname{E}[X ^{m]} = (| 2 \kappa \beta| ^{-m/\alpha})/(1+ \kappa (m)/(\alpha)) (\Gamma ( (1)/(- (m)/(2 \alpha) ) 2 \kappa))/(\Gamma ( 1+ \Gamma ( (1)/(+ (m)/(2 \alpha) ) 2 \kappa)) (m)/(\alpha) ) The median and the mode are:x _{rm{median} } (F_\kappa) = \beta^ \Bigg(\ln_\kappa (2)\Bigg)^ x _{rm{mode} } = \beta^ \Bigg(\frac\Bigg)^ \Bigg(\sqrt - 1 \Bigg)^ \quad (\alpha > 1)

Quantiles

The quantiles are given by the following expression

x _{rm{quantile} } (F_\kappa) = \beta^ \Bigg[\ln_\kappa \Bigg(\frac{1}{1 - F_\kappa} \Bigg) \Bigg]^

with 0 \leq F _{\kappa} \leq 1 .

Gini coefficient

The Gini coefficient is:^[3]

\operatorname{G} _{\kappa} = 1 - (\alpha + \kappa)/(\alpha + (1)/(2) \kappa) (\Gamma ( (1)/(\kappa) - (1)/(2 \alpha) ))/(\Gamma ( (1)/(\kappa) + (1)/(2 \alpha) )) (\Gamma ( (1)/(2 \kappa) + (1)/(2 \alpha) ))/(\Gamma ( (1)/(2 \kappa) - (1)/(2 \alpha) ))

Asymptotic behavior

The κ-Weibull distribution II behaves asymptotically as follows:^[3]

\lim _{x} f _{\kappa} (x) \sim (\alpha)/(\kappa) ( 2 \kappa \beta) ^{-1/\kappa} x ^{-1} \lim _{x \to 0 ^{+}} f _{\kappa} (x) = \alpha \beta x ^{\alpha}

Related distributions

• The κ-Weibull distribution is a generalization of:
  • κ-Exponential distribution of type II, when

\alpha = 1 ;

• • Exponential distribution when

\kappa = 0 and \alpha = 1 .

• A κ-Weibull distribution corresponds to a κ-deformed Rayleigh distribution when

\alpha = 2 and a Rayleigh distribution when \kappa = 0 and \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] and the interval distributions of seismic data, modeling extreme-event return intervals.^{[7] [8]}
• In epidemiology, the κ-Weibull distribution presents a universal feature for epidemiological analysis.^[9]

See also

• Giorgio Kaniadakis
• Kaniadakis statistics
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Gamma distribution
• Kaniadakis κ-Logistic distribution
• Kaniadakis κ-Erlang distribution

External links

• Kaniadakis Statistics on arXiv.org

Notes and References

1. Clementi . F. . Gallegati . M. . Kaniadakis . G. . 2007 . κ-generalized statistics in personal income distribution . The European Physical Journal B . en . 57 . 2 . 187–193 . 10.1140/epjb/e2007-00120-9 . physics/0607293 . 2007EPJB...57..187C . 15777288 . 1434-6028.
2. Clementi . F. . Di Matteo . T.. Tiziana Di Matteo (econophysicist) . 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 . en . 387 . 13 . 3201–3208 . 10.1016/j.physa.2008.01.109. 0710.3645 . 2590064 .
3. Kaniadakis . G. . 2021-01-01 . New power-law tailed distributions emerging in κ-statistics (a) . Europhysics Letters . 133 . 1 . 10002 . 10.1209/0295-5075/133/10002 . 2203.01743 . 2021EL....13310002K . 234144356 . 0295-5075.
4. Clementi . Fabio . Gallegati . Mauro . Kaniadakis . Giorgio . October 2010 . A model of personal income distribution with application to Italian data . Empirical Economics . en . 39 . 2 . 559–591 . 10.1007/s00181-009-0318-2 . 154273794 . 0377-7332.
5. 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 . 10.1088/1742-5468/2012/12/P12006 . 1209.4787 . 2012JSMTE..12..006C . 18961951 . 1742-5468.
6. da Silva . Sérgio Luiz E.F. . 2021 . κ -generalised Gutenberg–Richter law and the self-similarity of earthquakes . Chaos, Solitons & Fractals . en . 143 . 110622 . 10.1016/j.chaos.2020.110622. 2021CSF...14310622D . 234063959 .
7. 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 . en . 89 . 5 . 052142 . 10.1103/PhysRevE.89.052142 . 25353774 . 1308.1881 . 2014PhRvE..89e2142H . 22310350 . 1539-3755.
8. Hristopulos . Dionissios . Petrakis . Manolis . Kaniadakis . Giorgio . 2015-03-09 . Weakest-Link Scaling and Extreme Events in Finite-Sized Systems . Entropy . en . 17 . 3 . 1103–1122 . 10.3390/e17031103 . 2015Entrp..17.1103H . 1099-4300. free .
9. 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 . en . 10 . 1 . 19949 . 10.1038/s41598-020-76673-3 . 2045-2322 . 7673996 . 33203913. 2020NatSR..1019949K .