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_(x) = \frac \prod_^n \left 1 + (2m -n)\kappa \rightx^ \exp_\kappa(-x)

valid for $x \geq 0$ and $n = \textrm \,\,\textrm$, where $0 \leq , \kappa, < 1$ is the entropic index associated with the Kaniadakis entropy.

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

Cumulative distribution function

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

: F_\kappa(x) = \frac \prod_^n \left 1 + (2m -n)\kappa \right\int_0^x z^ \exp_\kappa(-z) dz

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

Survival distribution and hazard functions

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

S_\kappa(x) = 1 - \frac \prod_^n \left 1 + (2m -n)\kappa \right\int_0^x z^ \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:

$\frac = -h_\kappa S_\kappa(x)$

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 $0 \leq , \kappa, < 1$. Such members are determined from the ''κ''-Erlang cumulative distribution, which can be rewritten as:

: F_\kappa(x) = 1 - \left R_\kappa(x) + Q_\kappa(x) \sqrt \right\exp_\kappa(-x)

where

: $Q_\kappa(x) = N_\kappa \sum_^ \left( m + 1 \right) c_ x^m + \frac x^$
: $R_\kappa(x) = N_\kappa \sum_^ c_ x^m$

with

: N_\kappa = \frac \prod_^n \left 1 + (2m -n)\kappa \right
: $c_n = \frac$
: $c_ =0$
: $c_ = \frac$
: $c_m = \frac c_ \quad \textrm \quad 0 \leq m \leq n-3$

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:

: $f_(x) = (1 - \kappa^2) \exp_\kappa(-x)$
: $F_\kappa(x) = 1-\Big(\sqrt + \kappa^2 x \Big)\exp_k($

Second member

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

: $f_(x) = (1 - 4\kappa^2)\,x \,\exp_\kappa(-x)$
: $F_\kappa(x) = 1-\left(2\kappa^2 x^2 + 1 + x\sqrt \right) \exp_k($

Third member

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

: $f_(x) = \frac (1 - \kappa^2) (1 - 9\kappa^2)\,x^2 \,\exp_\kappa(-x)$
: $F_\kappa(x) = 1-\left\ \exp_\kappa(-x)$

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

References

External links

Kaniadakis Statistics on arXiv.org

{{DEFAULTSORT:Kaniadakis Erlang Distribution
Statistics
Probability distributions
Infinitely divisible probability distributions
Continuous distributions
Exponential family distributions