Kaniadakis Erlang distribution

The Kaniadakis κ-Erlang distribution 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 |issn=0295-5075|arxiv=2203.01743 |bibcode=2021EL....13310002K |s2cid=234144356 }}

: $$

f_{_{\kappa}}(x) = \frac{1}{ (n - 1)! } \prod_{m = 0}^n \left[ 1 + (2m -n)\kappa \right] x^{n - 1} \exp_\kappa(-x)

valid for $x\; \backslash geq\; 0$ and $n\; =\; \backslash textrm\{positive\}\; \backslash ,\backslash ,\backslash textrm\{integer\}$, where is the entropic index associated with the Kaniadakis entropy.

The ordinary Erlang Distribution is recovered as $\backslash kappa\; \backslash rightarrow\; 0$.

The cumulative distribution function of κ-Erlang distribution assumes the form:

: $F\_\backslash kappa(x)\; =\; \backslash frac\{1\}\{\; (n\; -\; 1)!\; \}\; \backslash prod\_\{m\; =\; 0\}^n\; \backslash left[\; 1\; +\; (2m\; -n)\backslash kappa\; \backslash right]\; \backslash int\_0^x\; z^\{n\; -\; 1\}\; \backslash exp\_\backslash kappa(-z)\; dz$

valid for $x\; \backslash geq\; 0$, where . The cumulative Erlang distribution is recovered in the classical limit $\backslash kappa\; \backslash rightarrow\; 0$.

The survival function of the κ-Erlang distribution is given by:

$S\_\backslash kappa(x)\; =\; 1\; -\; \backslash frac\{1\}\{\; (n\; -\; 1)!\; \}\; \backslash prod\_\{m\; =\; 0\}^n\; \backslash left[\; 1\; +\; (2m\; -n)\backslash kappa\; \backslash right]\; \backslash int\_0^x\; z^\{n\; -\; 1\}\; \backslash exp\_\backslash kappa(-z)\; dz$

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

A family of κ-distributions arises from the κ-Erlang distribution, each associated with a specific value of $n$, valid for $x\; \backslash ge\; 0$ and . Such members are determined from the κ-Erlang cumulative distribution, which can be rewritten as:

: $F\_\backslash kappa(x)\; =\; 1\; -\; \backslash left[\; R\_\backslash kappa(x)\; +\; Q\_\backslash kappa(x)\; \backslash sqrt\{1\; +\; \backslash kappa^2\; x^2\}\; \backslash right]\; \backslash exp\_\backslash kappa(-x)$

where

: $Q\_\backslash kappa(x)\; =\; N\_\backslash kappa\; \backslash sum\_\{m=0\}^\{n-3\}\; \backslash left(\; m\; +\; 1\; \backslash right)\; c\_\{m+1\}\; x^m\; +\; \backslash frac\{N\_\backslash kappa\}\{1-n^2\backslash kappa^2\}\; x^\{n-1\}$

: $R\_\backslash kappa(x)\; =\; N\_\backslash kappa\; \backslash sum\_\{m=0\}^\{n\}\; c\_\{m\}\; x^m$

with

: $N\_\backslash kappa\; =\; \backslash frac\{1\}\{\; (n\; -\; 1)!\; \}\; \backslash prod\_\{m\; =\; 0\}^n\; \backslash left[\; 1\; +\; (2m\; -n)\backslash kappa\; \backslash right]$

: $c\_n\; =\; \backslash frac\{\; n\backslash kappa^2\; \}\{\; 1\; -\; n^2\; \backslash kappa^2\}$

: $c\_\{n\; -\; 1\}\; =0$

: $c\_\{n\; -\; 2\}\; =\; \backslash frac\{\; n\; -\; 1\; \}\{\; (1\; -\; n^2\; \backslash kappa^2)\; [1\; -\; (n-2)^2\backslash kappa^2]\}$

: $c\_m\; =\; \backslash frac\{\; (m\; +\; 1)(m+2)\; \}\{\; 1\; -\; m^2\; \backslash kappa^2\}\; c\_\{m+2\}\; \backslash quad\; \backslash textrm\{for\}\; \backslash quad\; 0\; \backslash leq\; m\; \backslash leq\; n-3$

The first member ($n\; =\; 1$) of the κ-Erlang family is the κ-Exponential distribution of type I, in which the probability density function and the cumulative distribution function are defined as:

: $$

f_{_{\kappa}}(x) = (1 - \kappa^2) \exp_\kappa(-x)

: $F\_\backslash kappa(x)\; =\; 1-\backslash Big(\backslash sqrt\{1+\backslash kappa^2\; x^2\}\; +\; \backslash kappa^2\; x\; \backslash Big)\backslash exp\_k(\{-x)\}$

The second member ($n\; =\; 2$) of the κ-Erlang family has the probability density function and the cumulative distribution function defined as:

: $$

f_{_{\kappa}}(x) = (1 - 4\kappa^2)\,x \,\exp_\kappa(-x)

: $F\_\backslash kappa(x)\; =\; 1-\backslash left(2\backslash kappa^2\; x^2\; +\; 1\; +\; x\backslash sqrt\{1+\backslash kappa^2\; x^2\}\; \backslash right)\; \backslash exp\_k(\{-x)\}$

The second member ($n\; =\; 3$) has the probability density function and the cumulative distribution function defined as:

: $$

f_{_{\kappa}}(x) = \frac{1}{2} (1 - \kappa^2) (1 - 9\kappa^2)\,x^2 \,\exp_\kappa(-x)

: $F\_\backslash kappa(x)\; =\; 1-\backslash left\backslash \{\; \backslash frac\{3\}\{2\}\; \backslash kappa^2(1\; -\; \backslash kappa^2)x^3\; +\; x\; +\; \backslash left[\; 1\; +\; \backslash frac\{1\}\{2\}(1-\backslash kappa^2)x^2\; \backslash right]\; \backslash sqrt\{1+\backslash kappa^2\; x^2\}\backslash right\backslash \}\; \backslash exp\_\backslash kappa(-x)$

• The κ-Exponential distribution of type I is a particular case of the κ-Erlang distribution when $n\; =\; 1$;
• A κ-Erlang distribution corresponds to am ordinary exponential distribution when $\backslash kappa\; =\; 0$ and $n\; =\; 1$;

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

{{Reflist}}

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

{{ProbDistributions|continuous-semi-infinite}}

Category:Infinitely divisible probability distributions

Category:Exponential family distributions