Kaniadakis Erlang distribution

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 $\alpha =1$ and $\nu =n=$ positive integer. 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:

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

valid for $x\ge 0$ and $n=\text{positive}\text{integer}$, where $0\le |\kappa |<1$ is the entropic index associated with the Kaniadakis entropy.

The ordinary Erlang Distribution is recovered as $\kappa \to 0$.

Cumulative distribution function

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

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

valid for $x\ge 0$, where $0\le |\kappa |<1$. The cumulative Erlang distribution is recovered in the classical limit $\kappa \to 0$.

Survival distribution and hazard functions

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

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

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:

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

where

$h_{\kappa }$

is the hazard function.

Family distribution

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

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

where

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

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

with

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

$c_n=\frac{n{\kappa }^2}{1-n^2{\kappa }^2}$

$c_{n-1}=0$

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

$c_m=\frac{(m+1)(m+2)}{1-m^2{\kappa }^2}{c}_{m+2}\text{for}0\le m\le n-3$

First member

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:

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

$F_{\kappa }(x)=1-(\sqrt{1+{\kappa }^2{x}^2}+{\kappa }^{2}x){\exp }_k(-x)$

Second member

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

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

$F_{\kappa }(x)=1-(2{\kappa }^2{x}^2+1+x\sqrt{1+{\kappa }^2{x}^2}){\exp }_k(-x)$

Third member

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

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

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

Related distributions

• 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 $\kappa =0$ and $n=1$;

See also

• 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

External links

• Kaniadakis Statistics on arXiv.org