Kaniadakis Erlang distribution

Short description: Continuous probability distribution

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	[math]\displaystyle{ 0 \leq \kappa \lt 1 }[/math] [math]\displaystyle{ n = \textrm{positive} \,\,\textrm{integer} }[/math]
Support	[math]\displaystyle{ x \in [0, +\infty) }[/math]
PDF	[math]\displaystyle{ \prod_{m = 0}^n \left[ 1 + (2m -n)\kappa \right] \frac{ x^{n - 1} }{ (n - 1)! } \exp_\kappa(-x) }[/math]
CDF	[math]\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 }[/math]

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 [math]\displaystyle{ \alpha = 1 }[/math] and [math]\displaystyle{ \nu = n = }[/math] 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:

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

valid for [math]\displaystyle{ x \geq 0 }[/math] and [math]\displaystyle{ n = \textrm{positive} \,\,\textrm{integer} }[/math], where [math]\displaystyle{ 0 \leq |\kappa| \lt 1 }[/math] is the entropic index associated with the Kaniadakis entropy.

The ordinary Erlang Distribution is recovered as [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Cumulative distribution function

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

[math]\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 }[/math]

valid for [math]\displaystyle{ x \geq 0 }[/math], where [math]\displaystyle{ 0 \leq |\kappa| \lt 1 }[/math]. The cumulative Erlang distribution is recovered in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Survival distribution and hazard functions

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

[math]\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 }[/math]

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:

[math]\displaystyle{ \frac{ S_\kappa(x) }{ dx } = -h_\kappa S_\kappa(x) }[/math]

where [math]\displaystyle{ h_\kappa }[/math] is the hazard function.

Family distribution

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

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

where

[math]\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} }[/math]

[math]\displaystyle{ R_\kappa(x) = N_\kappa \sum_{m=0}^{n} c_{m} x^m }[/math]

with

[math]\displaystyle{ N_\kappa = \frac{1}{ (n - 1)! } \prod_{m = 0}^n \left[ 1 + (2m -n)\kappa \right] }[/math]

[math]\displaystyle{ c_n = \frac{ n\kappa^2 }{ 1 - n^2 \kappa^2} }[/math]

[math]\displaystyle{ c_{n - 1} =0 }[/math]

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

[math]\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 }[/math]

First member

The first member ([math]\displaystyle{ n = 1 }[/math]) 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:

[math]\displaystyle{ f_{_{\kappa}}(x) = (1 - \kappa^2) \exp_\kappa(-x) }[/math]

[math]\displaystyle{ F_\kappa(x) = 1-\Big(\sqrt{1+\kappa^2 x^2} + \kappa^2 x \Big)\exp_k({-x)} }[/math]

Second member

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

[math]\displaystyle{ f_{_{\kappa}}(x) = (1 - 4\kappa^2)\,x \,\exp_\kappa(-x) }[/math]

[math]\displaystyle{ F_\kappa(x) = 1-\left(2\kappa^2 x^2 + 1 + x\sqrt{1+\kappa^2 x^2} \right) \exp_k({-x)} }[/math]

Third member

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

[math]\displaystyle{ f_{_{\kappa}}(x) = \frac{1}{2} (1 - \kappa^2) (1 - 9\kappa^2)\,x^2 \,\exp_\kappa(-x) }[/math]

[math]\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) }[/math]

Related distributions

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

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