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 and 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:
valid for and , where is the entropic index associated with the Kaniadakis entropy.
The ordinary Erlang Distribution is recovered as .
Cumulative distribution function The cumulative distribution function of κ-Erlang distribution assumes the form:
valid for , where . The cumulative Erlang distribution is recovered in the classical limit .
Survival distribution and hazard functions The survival function of the κ-Erlang distribution is given by:
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: where is the hazard function.
Family distribution A family of κ-distributions arises from the κ-Erlang distribution, each associated with a specific value of , valid for and . Such members are determined from the κ-Erlang cumulative distribution, which can be rewritten as:
where
with
First member The first member () 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:
Second member The second member () of the κ-Erlang family has the probability density function and the cumulative distribution function defined as:
Third member The second member () has the probability density function and the cumulative distribution function defined as:
Related distributions
The κ-Exponential distribution of type I is a particular case of the κ-Erlang distribution when ; A κ-Erlang distribution corresponds to am ordinary exponential distribution when and ;
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
References
External links Kaniadakis Statistics on arXiv.org