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 ${\displaystyle \alpha =1}${\displaystyle \alpha =1} and ${\displaystyle \nu =n=}${\displaystyle \nu =n=} positive integer.^[1] 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.

The Kaniadakis κ-Erlang distribution has the following probability density function:^[1]

${\displaystyle f_{_{\kappa }}(x)={\frac {1}{(n-1)!}}\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]x^{n-1}\exp _{\kappa }(-x)}${\displaystyle 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 ${\displaystyle x\geq 0}${\displaystyle x\geq 0} and ${\displaystyle n={\textrm {positive}},円,円{\textrm {integer}}}${\displaystyle n={\textrm {positive}},円,円{\textrm {integer}}}, where ${\displaystyle 0\leq |\kappa |<1}${\displaystyle 0\leq |\kappa |<1} is the entropic index associated with the Kaniadakis entropy.

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

The cumulative distribution function of κ-Erlang distribution assumes the form:^[1]

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

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

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

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

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:

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

where ${\displaystyle h_{\kappa }}${\displaystyle h_{\kappa }} is the hazard function.

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

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

where

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

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

with

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

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

${\displaystyle c_{n-1}=0}${\displaystyle c_{n-1}=0}

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

${\displaystyle c_{m}={\frac {(m+1)(m+2)}{1-m^{2}\kappa ^{2}}}c_{m+2}\quad {\textrm {for}}\quad 0\leq m\leq n-3}${\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}

The first member (${\displaystyle n=1}${\displaystyle 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:

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

${\displaystyle F_{\kappa }(x)=1-{\Big (}{\sqrt {1+\kappa ^{2}x^{2}}}+\kappa ^{2}x{\Big )}\exp _{k}({-x)}}${\displaystyle F_{\kappa }(x)=1-{\Big (}{\sqrt {1+\kappa ^{2}x^{2}}}+\kappa ^{2}x{\Big )}\exp _{k}({-x)}}

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

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

${\displaystyle F_{\kappa }(x)=1-\left(2\kappa ^{2}x^{2}+1+x{\sqrt {1+\kappa ^{2}x^{2}}}\right)\exp _{k}({-x)}}${\displaystyle F_{\kappa }(x)=1-\left(2\kappa ^{2}x^{2}+1+x{\sqrt {1+\kappa ^{2}x^{2}}}\right)\exp _{k}({-x)}}

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

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

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

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

• 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

• Kaniadakis Statistics on arXiv.org

• Infinitely divisible probability distributions
• Exponential family distributions

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