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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.

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Type I

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Probability density function

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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 .

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Type II

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Probability density function

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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]

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Applications

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

See also

References

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