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 α = 1 {\displaystyle \alpha =1} and ν = n = {\displaystyle \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 κ ( x ) = 1 ( n − 1 ) ! ∏ m = 0 n [ 1 + ( 2 m − n ) κ ] x n − 1 exp κ ( − 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 x ≥ 0 {\displaystyle x\geq 0} and n = positive integer {\displaystyle n={\textrm {positive}}\,\,{\textrm {integer}}} , where 0 ≤ | κ | < 1 {\displaystyle 0\leq |\kappa |<1} is the entropic index associated with the Kaniadakis entropy. The ordinary Erlang Distribution is recovered as κ → 0 {\displaystyle \kappa \rightarrow 0} .
Cumulative distribution function The cumulative distribution function of κ-Erlang distribution assumes the form:
F κ ( x ) = 1 ( n − 1 ) ! ∏ m = 0 n [ 1 + ( 2 m − n ) κ ] ∫ 0 x z n − 1 exp κ ( − z ) d z {\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 x ≥ 0 {\displaystyle x\geq 0} , where 0 ≤ | κ | < 1 {\displaystyle 0\leq |\kappa |<1} . The cumulative Erlang distribution is recovered in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0} .
Survival distribution and hazard functions The survival function of the κ-Erlang distribution is given by:
S κ ( x ) = 1 − 1 ( n − 1 ) ! ∏ m = 0 n [ 1 + ( 2 m − n ) κ ] ∫ 0 x z n − 1 exp κ ( − z ) d z {\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: S κ ( x ) d x = − h κ S κ ( x ) {\displaystyle {\frac {S_{\kappa }(x)}{dx}}=-h_{\kappa }S_{\kappa }(x)} where h κ {\displaystyle h_{\kappa }} is the hazard function.
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