κ-Erlang distribution

Probability density function Plot of the κ-Erlang distribution for typical κ-values and n=1, 2,3. The case κ=0 corresponds to the ordinary Erlang distribution.
Parameters	${\displaystyle 0\leq \kappa <1}$ ${\displaystyle n={\textrm {positive))\,\,{\textrm {integer))}$
Support	${\displaystyle x\in [0,+\infty )}$
PDF	${\displaystyle \prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]{\frac {x^{n-1)){(n-1)!))\exp _{\kappa }(-x)}$
CDF	${\displaystyle {\frac {1}{(n-1)!))\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]\int _{0}^{x}z^{n-1}\exp _{\kappa }(-z)dz}$

The Kaniadakis Erlang distribution (or κ-Erlang Gamma distribution) is a family of continuous statistical distributions, which is a particular case of the κ-Gamma distribution, when ${\displaystyle \alpha =1}$ and ${\displaystyle \nu =n=}$ positive integer.^[1] The first member of this family is the κ-exponential distribution of Type I. The κ-Erlang is a κ-deformed version of the Erlang distribution. It is one example of a Kaniadakis distribution.

Characterization

Probability density function

The Kaniadakis κ-Erlang distribution has the following probability density function:^[1]

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

valid for ${\displaystyle x\geq 0}$ and ${\displaystyle n={\textrm {positive))\,\,{\textrm {integer))}$, where ${\displaystyle 0\leq |\kappa |<1}$ is the entropic index associated with the Kaniadakis entropy.

The ordinary Erlang Distribution is recovered as ${\displaystyle \kappa \rightarrow 0}$.

Cumulative distribution function

The cumulative distribution function of κ-Erlang distribution assumes the form:^[1]

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

valid for ${\displaystyle x\geq 0}$, where ${\displaystyle 0\leq |\kappa |<1}$. The cumulative Erlang distribution is recovered in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

Survival distribution and hazard functions

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

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

The survival function of the κ-Erlang distribution enables the determination of hazard functions in closed form through the solution of the κ-rate equation:

${\displaystyle {\frac {S_{\kappa }(x)}{dx))=-h_{\kappa }S_{\kappa }(x)}$

where ${\displaystyle h_{\kappa ))$ is the hazard function.

Family distribution

A family of κ-distributions arises from the κ-Erlang distribution, each associated with a specific value of ${\displaystyle n}$, valid for ${\displaystyle x\geq 0}$ and ${\displaystyle 0\leq |\kappa |<1}$. Such members are determined from the κ-Erlang cumulative distribution, which can be rewritten as:

${\displaystyle F_{\kappa }(x)=1-\left[R_{\kappa }(x)+Q_{\kappa }(x){\sqrt {1+\kappa ^{2}x^{2))}\right]\exp _{\kappa }(-x)}$

where

${\displaystyle Q_{\kappa }(x)=N_{\kappa }\sum _{m=0}^{n-3}\left(m+1\right)c_{m+1}x^{m}+{\frac {N_{\kappa )){1-n^{2}\kappa ^{2))}x^{n-1))$

${\displaystyle R_{\kappa }(x)=N_{\kappa }\sum _{m=0}^{n}c_{m}x^{m))$

with

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

${\displaystyle c_{n}={\frac {n\kappa ^{2)){1-n^{2}\kappa ^{2))))$

${\displaystyle c_{n-1}=0}$

${\displaystyle c_{n-2}={\frac {n-1}{(1-n^{2}\kappa ^{2})[1-(n-2)^{2}\kappa ^{2}]))}$

${\displaystyle c_{m}={\frac {(m+1)(m+2)}{1-m^{2}\kappa ^{2))}c_{m+2}\quad {\textrm {for))\quad 0\leq m\leq n-3}$

First member

The first member (${\displaystyle n=1}$) 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:

${\displaystyle f_{_{\kappa ))(x)=(1-\kappa ^{2})\exp _{\kappa }(-x)}$

${\displaystyle F_{\kappa }(x)=1-{\Big (}{\sqrt {1+\kappa ^{2}x^{2))}+\kappa ^{2}x{\Big )}\exp _{k}({-x)))$

Second member

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

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

${\displaystyle F_{\kappa }(x)=1-\left(2\kappa ^{2}x^{2}+1+x{\sqrt {1+\kappa ^{2}x^{2))}\right)\exp _{k}({-x)))$

Third member

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

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

${\displaystyle F_{\kappa }(x)=1-\left\((\frac {3}{2))\kappa ^{2}(1-\kappa ^{2})x^{3}+x+\left[1+{\frac {1}{2))(1-\kappa ^{2})x^{2}\right]{\sqrt {1+\kappa ^{2}x^{2))}\right\}\exp _{\kappa }(-x)}$

Related distributions

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