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.

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 \left[ 1 + (2m -n)\kappa \right] x^{n - 1} \exp_\kappa(-x) $$

valid for $$x \geq 0$$ and $$n = \textrm{positive} \,\,\textrm{integer} $$, 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:


 * $$F_\kappa(x) = \frac{1}{ (n - 1)! } \prod_{m = 0}^n \left[ 1 + (2m -n)\kappa \right] \int_0^x z^{n - 1} \exp_\kappa(-z) dz $$

valid for $$x \geq 0$$, 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{1}{ (n - 1)! } \prod_{m = 0}^n \left[ 1 + (2m -n)\kappa \right] \int_0^x z^{n - 1} \exp_\kappa(-z) dz $"The survival function of the κ-Erlang distribution enables the determination of hazard functions in closed form through the solution of the κ-rate equation:"$\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 $$0 \leq |\kappa| < 1$$. Such members are determined from the κ-Erlang cumulative distribution, which can be rewritten as:


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

where


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

with


 * $$N_\kappa = \frac{1}{ (n - 1)! } \prod_{m = 0}^n \left[ 1 + (2m -n)\kappa \right] $$
 * $$c_n = \frac{ n\kappa^2 }{ 1 - n^2 \kappa^2} $$
 * $$c_{n - 1} =0 $$
 * $$c_{n - 2} = \frac{ n - 1 }{ (1 - n^2 \kappa^2) [1 - (n-2)^2\kappa^2]} $$
 * $$c_m = \frac{ (m + 1)(m+2) }{ 1 - m^2 \kappa^2} c_{m+2} \quad \textrm{for} \quad 0 \leq m \leq n-3 $$

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_{_{\kappa}}(x) = (1 - \kappa^2) \exp_\kappa(-x) $$
 * $$F_\kappa(x) = 1-\Big(\sqrt{1+\kappa^2 x^2} + \kappa^2 x \Big)\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-\left(2\kappa^2 x^2 + 1 + x\sqrt{1+\kappa^2 x^2} \right) \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-\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 $$n = 1$$;
 * A κ-Erlang distribution corresponds to am ordinary exponential distribution when $$\kappa = 0$$ and $$n = 1$$;