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. 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:orThe 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:orThe skewness of the ordinary exponential distribution is recovered in the limit .
Quantiles
The quantiles are given by the following expressionwith , 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. See also Giorgio Kaniadakis Kaniadakis statistics Kaniadakis distribution Kaniadakis κ-Gaussian distribution Kaniadakis κ-Gamma distribution Kaniadakis κ-Weibull distribution Kaniadakis κ-Logistic distribution Kaniadakis κ-Erlang distribution Exponential distribution
References
External links Giorgio Kaniadakis Google Scholar page Kaniadakis Statistics on arXiv.org