The Kaniadakis exponential distribution (or κ-exponential distribution) is a probability distribution arising from the maximization of the Kaniadakis entropy under appropriate constraints.

It is one example of a Kaniadakis distribution.

The κ-exponential is a generalization of the exponential distribution in the same way that Kaniadakis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy.

[1] The κ-exponential distribution of Type I is a particular case of the κ-Gamma distribution, whilst the κ-exponential distribution of Type II is a particular case of the κ-Weibull distribution.

The Kaniadakis κ-exponential distribution of Type I is part of a class of statistical distributions emerging from the Kaniadakis κ-statistics which exhibit power-law tails.

This distribution has the following probability density function:[2] valid for

is the entropic index associated with the Kaniadakis entropy and

The exponential distribution is recovered as

The cumulative distribution function of κ-exponential distribution of Type I is given by for

The cumulative exponential distribution is recovered in the classical limit

The κ-exponential distribution of type I has moment of order

The expectation is defined as: and the variance is: The kurtosis of the κ-exponential distribution of type I may be computed thought:

Thus, the kurtosis of the κ-exponential distribution of type I distribution is given by:

The kurtosis of the ordinary exponential distribution is recovered in the limit

The skewness of the κ-exponential distribution of type I may be computed thought:

Thus, the skewness of the κ-exponential distribution of type I distribution is given by:

The kurtosis of the ordinary exponential distribution is recovered in the limit

The Kaniadakis κ-exponential distribution of Type II also is part of a class of statistical distributions emerging from the Kaniadakis κ-statistics which exhibit power-law tails, but with different constraints.

This distribution is a particular case of the Kaniadakis κ-Weibull distribution with

is the entropic index associated with the Kaniadakis entropy and

is known as rate parameter.

The exponential distribution is recovered as

The cumulative distribution function of κ-exponential distribution of Type II is given by for

The cumulative exponential distribution is recovered in the classical limit

The κ-exponential distribution of type II has moment of order

given by[2] The expectation value and the variance are: The mode is given by: The kurtosis of the κ-exponential distribution of type II may be computed thought: Thus, the kurtosis of the κ-exponential distribution of type II distribution is given by: or The skewness of the κ-exponential distribution of type II may be computed thought:

Thus, the skewness of the κ-exponential distribution of type II distribution is given by:

The skewness of the ordinary exponential distribution is recovered in the limit

The Lorenz curve associated with the κ-exponential distribution of type II is given by:[2]

The κ-exponential distribution of type II behaves asymptotically as follows:[2] The κ-exponential distribution has been applied in several areas, such as: