Kaniadakis Erlang distribution explained

κ-Erlang distribution
Type:	density
Parameters:	0 \leq \kappa < 1 n = rm{positive} rm{integer}
Support:	x \in [ 0 , +inf ty) |pdf=\prod ^{n} _{m =0} \left[ 1 + ( 2 m - n)\kappa \right] (x ^{n})/((n -1 )!) \exp _{\kappa(-x)} |cdf=(1)/((n -1 )!) \prod ^{n} _{m =0} \left[ 1 + ( 2 m - n)\kappa \right] \int ^{x} _{0} z ^{n} \exp _{\kappa(-z)} dz |pdf_image=FILE:Kaniadakis Erlang Distribution pdf.png|pdf_caption=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.

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:^[1]

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

valid for

x \geq 0 and n = rm{positive} rm{integer} , where 0 \leq |\kappa| < 1 is the entropic index associated with the Kaniadakis entropy.

The ordinary Erlang Distribution is recovered as

\kappa → 0 .

Cumulative distribution function

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

F _{\kappa(x)} = (1)/((n -1 )!) \prod ^{n} _{m =0} \left[ 1 + ( 2 m - n)\kappa \right] \int ^{x} _{0} z ^{n} \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 → 0 .

Survival distribution and hazard functions

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

S _{\kappa(x)} = 1 - (1)/((n -1 )!) \prod ^{n} _{m =0} \left[ 1 + ( 2 m - n)\kappa \right] \int ^{x} _{0} z ^{n} \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:

(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 ^{n -3} _{m =0} \left( m + 1 \right) c _{m+1} x ^{m} + (N _{\kappa})/(1- n ^{2\kappa} ^{2}) x ^{n-1} R _{\kappa(x)} = N _{\kappa} \sum ^{n} _{m =0} c _{m} x ^{m}

with

N _{\kappa} = (1)/((n -1 )!) \prod ^{n} _{m =0} \left[ 1 + ( 2 m - n)\kappa \right] c _{n} = (n\kappa ^{2})/(1- n ^{2} \kappa ^{2}) c _{n} =0 c _{n} = (n -1)/(( 1- n ^{2} \kappa ^{2)} [ 1- (n -2 ) ^{2\kappa} ^{2]}) c _{m} = ((m +1 )(m +2 ))/(1- m ^{2} \kappa ^{2}) c _{m+2} rm{for} 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- ( \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- \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) = (1)/(2) ( 1 - \kappa ^{2)} ( 1 - 9 \kappa ^{2)x} ^{2} \exp _{\kappa(-x)} F _{\kappa(x)} = 1- \left\{ (3)/(2) \kappa ^{2(1} - \kappa ^{2)x} ^{3} + x + \left[ 1 + (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 ;

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

Notes and References

1. Kaniadakis . G. . 2021-01-01 . New power-law tailed distributions emerging in κ-statistics (a) . Europhysics Letters . 133 . 1 . 10002 . 10.1209/0295-5075/133/10002 . 0295-5075. 2203.01743 . 2021EL....13310002K . 234144356 .