The Kaniadakis Erlang distribution (or ú-Erlang Gamma distribution) is a family of continuous statistical distributions, which is a particular case of the ú-Gamma distribution, when and positive integer. The first member of this family is the ú-exponential distribution of Type I. The ú-Erlang is a ú-deformed version of the Erlang distribution. It is one example of a Kaniadakis distribution.
The Kaniadakis ú-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 .
The cumulative distribution function of ú-Erlang distribution assumes the form:
valid for , where . The cumulative Erlang distribution is recovered in the classical limit .
The survival function of the ú-Erlang distribution is given by:<blockquote></blockquote>The survival function of the ú-Erlang distribution enables the determination ofàhazard functions in closed form through the solution of the ú-rate equation:<blockquote></blockquote>where is the hazard function.
A family of ú-distributions arises from the ú-Erlang distribution, each associated with a specific value of , valid for and . Such members are determined from the ú-Erlang cumulative distribution, which can be rewritten as:
where
with
The first member () of the ú-Erlang family is the ú-Exponential distribution of type I, in which the probability density function and the cumulative distribution function are defined as:
The second member () of the ú-Erlang family has the probability density function and the cumulative distribution function defined as:
The second member () has the probability density function and the cumulative distribution function defined as: