Kaniadakis exponential distribution
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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.
Type I
Summarize
Perspective
Probability density function
Probability density function ![]() | |||
Cumulative distribution function ![]() | |||
Parameters |
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Variance | |||
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Excess kurtosis | |||
Method of moments |
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 , 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[2]
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:
or
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
Summarize
Perspective
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:[2]
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[2]
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:
or
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:[2]
The Gini coefficient is
Asymptotic behavior
The κ-exponential distribution of type II behaves asymptotically as follows:[2]
Applications
The κ-exponential distribution has been applied in several areas, such as:
- In geomechanics, for analyzing the properties of rock masses;[3]
- In quantum theory, in physical analysis using Planck's radiation law;[4]
- In inverse problems, the κ-exponential distribution has been used to formulate a robust approach;[5]
- In Network theory.[6]
See also
References
External links
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