Kaniadakis Weibull distribution

The Kaniadakis κ-Weibull distribution is exhibits power-law right tails, and it has the following probability density function:{{Cite journal |last=Kaniadakis |first=G. |date=2021-01-01 |title=New power-law tailed distributions emerging in κ-statistics (a) |url=https://iopscience.iop.org/article/10.1209/0295-5075/133/10002 |journal=Europhysics Letters |volume=133 |issue=1 |pages=10002 |doi=10.1209/0295-5075/133/10002 |arxiv=2203.01743 |bibcode=2021EL....13310002K |s2cid=234144356 |issn=0295-5075}}

: $$

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 $x\; \backslash geq\; 0$, where is the entropic index associated with the Kaniadakis entropy, $\backslash beta\; >\; 0$ is the scale parameter, and $\backslash alpha\; >\; 0$ is the shape parameter or Weibull modulus.

The Weibull distribution is recovered as $\backslash kappa\; \backslash rightarrow\; 0.$

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

$F\_\backslash kappa(x)\; =$

1 - \exp_\kappa(-\beta x^\alpha)

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

:$S\_\backslash kappa(x)\; =\; \backslash exp\_\backslash kappa(-\backslash beta\; x^\backslash alpha)$

valid for $x\; \backslash geq\; 0$. The survival Weibull distribution is recovered in the classical limit $\backslash kappa\; \backslash rightarrow\; 0$.

File:Kaniadakisweibull1.png

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

$\backslash frac\{\; S\_\backslash kappa(x)\; \}\{\; dx\; \}\; =\; -h\_\backslash kappa\; S\_\backslash kappa(x)$

:$h\_\backslash kappa\; =\; \backslash frac\{\backslash alpha\; \backslash beta\; x^\{\backslash alpha-1\}\}\{\backslash sqrt\{1+\backslash kappa^2\; \backslash beta^2\; x^\{2\backslash alpha\}\; \}\}$

The cumulative κ-Weibull distribution is related to the κ-hazard function by the following expression:

:$S\_\backslash kappa\; =\; e^\{-H\_\backslash kappa(x)\}$

where

:$H\_\backslash kappa\; (x)\; =\; \backslash int\_0^x\; h\_\backslash kappa(z)\; dz$

:$H\_\backslash kappa\; (x)\; =\; \backslash frac\{1\}\{\backslash kappa\}\; \backslash textrm\{arcsinh\}\backslash left(\backslash kappa\; \backslash beta\; x^\backslash alpha\; \backslash right)$

is the cumulative κ-hazard function. The cumulative hazard function of the Weibull distribution is recovered in the classical limit $\backslash kappa\; \backslash rightarrow\; 0$: $H(x)\; =\; \backslash beta\; x^\backslash alpha$ .

The κ-Weibull distribution has moment of order $m\; \backslash in\; \backslash mathbb\{N\}$ given by

:$\backslash operatorname\{E\}[X^m]\; =\; \backslash frac\{|2\backslash kappa\; \backslash beta|^\{-m/\backslash alpha\}\}\{1+\backslash kappa\; \backslash frac\{m\}\{\backslash alpha\}\}\; \backslash frac\{\backslash Gamma\backslash Big(\backslash frac\{1\}\{2\backslash kappa\}-\backslash frac\{m\}\{2\backslash alpha\}\backslash Big)\}\{\backslash Gamma\backslash Big(\backslash frac\{1\}\{2\backslash kappa\}+\backslash frac\{m\}\{2\backslash alpha\}\backslash Big)\}\; \backslash Gamma\backslash Big(1+\backslash frac\{m\}\{\backslash alpha\}\backslash Big)$

The median and the mode are:

: $x\_\{\backslash textrm\{median\}\}\; (F\_\backslash kappa)\; =\; \backslash beta^\{-1/\backslash alpha\}\; \backslash Bigg(\backslash ln\_\backslash kappa\; (2)\backslash Bigg)^\{1/\backslash alpha\}$

: $x\_\{\backslash textrm\{mode\}\}\; =\; \backslash beta^\{\; -1\; /\; \backslash alpha\; \}\; \backslash Bigg(\; \backslash frac\{\; \backslash alpha^2\; +\; 2\; \backslash kappa^2\; (\backslash alpha\; -\; 1\; )\}\{\; 2\; \backslash kappa^2\; (\; \backslash alpha^2\; -\; \backslash kappa^2)\}\backslash Bigg)^\{1/2\; \backslash alpha\}\; \backslash Bigg(\; \backslash sqrt\{1\; +\; \backslash frac\{4\; \backslash kappa^2\; (\backslash alpha^2\; -\; \backslash kappa^2\; )(\; \backslash alpha\; -\; 1)^2\}\{\; [\; \backslash alpha^2\; +\; 2\; \backslash kappa^2\; (\backslash alpha\; -\; 1)\; ]^2\}\; \}\; -\; 1\; \backslash Bigg)^\{1/2\; \backslash alpha\}\; \backslash quad\; (\backslash alpha\; >\; 1)$

The quantiles are given by the following expression

$x\_\{\backslash textrm\{quantile\}\}\; (F\_\backslash kappa)\; =\; \backslash beta^\{-1\; /\; \backslash alpha\; \}\; \backslash Bigg[\; \backslash ln\_\backslash kappa\; \backslash Bigg(\backslash frac\{1\}\{1\; -\; F\_\backslash kappa\}\; \backslash Bigg)\; \backslash Bigg]^\{1/\; \backslash alpha\}$

The Gini coefficient is:

$\backslash operatorname\{G\}\_\backslash kappa\; =\; 1\; -\; \backslash frac\{\backslash alpha\; +\; \backslash kappa\}\{\; \backslash alpha\; +\; \backslash frac\{1\}\{2\}\backslash kappa\; \}\; \backslash frac\{\backslash Gamma\backslash Big(\; \backslash frac\{1\}\{\backslash kappa\}\; -\; \backslash frac\{1\}\{2\; \backslash alpha\}\backslash Big)\}\{\backslash Gamma\backslash Big(\; \backslash frac\{1\}\{\backslash kappa\}\; +\; \backslash frac\{1\}\{2\; \backslash alpha\}\backslash Big)\}\; \backslash frac\{\backslash Gamma\backslash Big(\; \backslash frac\{1\}\{2\; \backslash kappa\}\; +\; \backslash frac\{1\}\{2\; \backslash alpha\}\backslash Big)\}\{\backslash Gamma\backslash Big(\; \backslash frac\{1\}\{\; 2\backslash kappa\}\; -\; \backslash frac\{1\}\{2\; \backslash alpha\}\backslash Big)\}$

The κ-Weibull distribution II behaves asymptotically as follows:

: $\backslash lim\_\{x\; \backslash to\; +\backslash infty\}\; f\_\backslash kappa\; (x)\; \backslash sim\; \backslash frac\{\backslash alpha\}\{\backslash kappa\}\; (2\; \backslash kappa\; \backslash beta)^\{-1/\backslash kappa\}\; x^\{-1\; -\; \backslash alpha/\backslash kappa\}$

: $\backslash lim\_\{x\; \backslash to\; 0^+\}\; f\_\backslash kappa\; (x)\; =\; \backslash alpha\; \backslash beta\; x^\{\backslash alpha\; -\; 1\}$

• The κ-Weibull distribution is a generalization of:
• κ-Exponential distribution of type II, when $\backslash alpha\; =\; 1$;
• Exponential distribution when $\backslash kappa\; =\; 0$ and $\backslash alpha\; =\; 1$.
• A κ-Weibull distribution corresponds to a κ-deformed Rayleigh distribution when $\backslash alpha\; =\; 2$ and a Rayleigh distribution when $\backslash kappa\; =\; 0$ and $\backslash alpha\; =\; 2$.

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.{{Cite journal |last1=Clementi |first1=Fabio |last2=Gallegati |first2=Mauro |last3=Kaniadakis |first3=Giorgio |date=October 2010 |title=A model of personal income distribution with application to Italian data |url=http://link.springer.com/10.1007/s00181-009-0318-2 |journal=Empirical Economics |language=en |volume=39 |issue=2 |pages=559–591 |doi=10.1007/s00181-009-0318-2 |s2cid=154273794 |issn=0377-7332}}{{Cite journal |last1=Clementi |first1=F |last2=Gallegati |first2=M |last3=Kaniadakis |first3=G |date=2012-12-06 |title=A generalized statistical model for the size distribution of wealth |url=https://iopscience.iop.org/article/10.1088/1742-5468/2012/12/P12006 |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=2012 |issue=12 |pages=P12006 |doi=10.1088/1742-5468/2012/12/P12006 |arxiv=1209.4787 |bibcode=2012JSMTE..12..006C |s2cid=18961951 |issn=1742-5468}}
• In seismology, the κ-Weibull represents the statistical distribution of magnitude of the earthquakes distributed across the Earth, generalizing the Gutenberg–Richter law,{{Cite journal |last=da Silva |first=Sérgio Luiz E.F. |date=2021 |title=κ -generalised Gutenberg–Richter law and the self-similarity of earthquakes |url=https://linkinghub.elsevier.com/retrieve/pii/S0960077920310134 |journal=Chaos, Solitons & Fractals |language=en |volume=143 |pages=110622 |doi=10.1016/j.chaos.2020.110622|bibcode=2021CSF...14310622D |s2cid=234063959 }} and the interval distributions of seismic data, modeling extreme-event return intervals.{{Cite journal |last1=Hristopulos |first1=Dionissios T. |last2=Petrakis |first2=Manolis P. |last3=Kaniadakis |first3=Giorgio |date=2014-05-28 |title=Finite-size effects on return interval distributions for weakest-link-scaling systems |url=https://link.aps.org/doi/10.1103/PhysRevE.89.052142 |journal=Physical Review E |language=en |volume=89 |issue=5 |pages=052142 |doi=10.1103/PhysRevE.89.052142 |pmid=25353774 |arxiv=1308.1881 |bibcode=2014PhRvE..89e2142H |s2cid=22310350 |issn=1539-3755}}{{Cite journal |last1=Hristopulos |first1=Dionissios |last2=Petrakis |first2=Manolis |last3=Kaniadakis |first3=Giorgio |date=2015-03-09 |title=Weakest-Link Scaling and Extreme Events in Finite-Sized Systems |journal=Entropy |language=en |volume=17 |issue=3 |pages=1103–1122 |doi=10.3390/e17031103 |bibcode=2015Entrp..17.1103H |issn=1099-4300|doi-access=free }}
• In epidemiology, the κ-Weibull distribution presents a universal feature for epidemiological analysis.{{Cite journal |last1=Kaniadakis |first1=Giorgio |last2=Baldi |first2=Mauro M. |last3=Deisboeck |first3=Thomas S. |last4=Grisolia |first4=Giulia |last5=Hristopulos |first5=Dionissios T. |last6=Scarfone |first6=Antonio M. |last7=Sparavigna |first7=Amelia |last8=Wada |first8=Tatsuaki |last9=Lucia |first9=Umberto |date=2020 |title=The κ-statistics approach to epidemiology |journal=Scientific Reports |language=en |volume=10 |issue=1 |pages=19949 |doi=10.1038/s41598-020-76673-3 |issn=2045-2322 |pmc=7673996 |pmid=33203913|bibcode=2020NatSR..1019949K }}

• 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

{{Reflist}}

• [https://arxiv.org/search/?query=kaniadakis+statistics&searchtype=all&abstracts=show&order=-announced_date_first&size=200 Kaniadakis Statistics on arXiv.org]

Category:Continuous distributions

Category:Exponential family distributions

Category:Survival analysis