Kaniadakis exponential distribution

The Kaniadakis 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. 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.

Type I

Probability density function

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:
valid for, where is the entropic index associated with the Kaniadakis entropy and is known as rate parameter. The exponential distribution is recovered as

Cumulative distribution function

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.

Properties

Moments, expectation value and variance

The κ-exponential distribution of type I has moment of order given by
where is finite if.
The expectation is defined as:
and the variance is:

Kurtosis

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.

Skewness

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.

Type II

Probability density function

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:
valid for, where is the entropic index associated with the Kaniadakis entropy and is known as rate parameter.
The exponential distribution is recovered as

Cumulative distribution function

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.

Properties

Moments, expectation value and variance

The κ-exponential distribution of type II has moment of order given by
The expectation value and the variance are:
The mode is given by:

Kurtosis

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

Skewness

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.

Quantiles

The quantiles are given by the following expression

with, in which the median is the case :

Lorenz curve

The Lorenz curve associated with the κ-exponential distribution of type II is given by:
The Gini coefficient is

Asymptotic behavior

The κ-exponential distribution of type II behaves asymptotically as follows:

Applications

The κ-exponential distribution has been applied in several areas, such as:

In geomechanics, for analyzing the properties of rock masses;
In quantum theory, in physical analysis using Planck's radiation law;
In inverse problems, the κ-exponential distribution has been used to formulate a robust approach;
In Network theory.