The Kaniadakis exponential distribution (or ú-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 ú-exponential is a generalization of the exponential distribution in the same way that Kaniadakis entropy is a generalization of standard BoltzmannâÂÂGibbs entropy or Shannon entropy. The ú-exponential distribution of Type I is a particular case of the ú-Gamma distribution, whilst the ú-exponential distribution of Type II is a particular case of the ú-Weibull distribution.
The Kaniadakis ú-exponential distribution of Type I is part of a class of statistical distributions emerging from the Kaniadakis ú-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
The cumulative distribution function of ú-exponential distribution of Type I is given by
for . The cumulative exponential distribution is recovered in the classical limit .
The ú-exponential distribution of type I has moment of order given by
where is finite if .
The expectation is defined as:
and the variance is:
The kurtosis of the ú-exponential distribution of type I may be computed thought:
Thus, the kurtosis of the ú-exponential distribution of type I distribution is given by:<blockquote></blockquote>or<blockquote></blockquote>The kurtosis of the ordinary exponential distribution is recovered in the limit .
The skewness of the ú-exponential distribution of type I may be computed thought:
Thus, the skewness of the ú-exponential distribution of type I distribution is given by:<blockquote></blockquote>The kurtosis of the ordinary exponential distribution is recovered in the limit .
The Kaniadakis ú-exponential distribution of Type II also is part of a class of statistical distributions emerging from the Kaniadakis ú-statistics which exhibit power-law tails, but with different constraints. This distribution is a particular case of the Kaniadakis ú-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
The cumulative distribution function of ú-exponential distribution of Type II is given by
for . The cumulative exponential distribution is recovered in the classical limit .
The ú-exponential distribution of type II has moment of order given by
The expectation value and the variance are:
The mode is given by:
The kurtosis of the ú-exponential distribution of type II may be computed thought:
Thus, the kurtosis of the ú-exponential distribution of type II distribution is given by:
or
The skewness of the ú-exponential distribution of type II may be computed thought:
Thus, the skewness of the ú-exponential distribution of type II distribution is given by:<blockquote></blockquote>or<blockquote></blockquote>The skewness of the ordinary exponential distribution is recovered in the limit .
The quantiles are given by the following expression<blockquote></blockquote>with , in which the median is the case :<blockquote></blockquote>
The Lorenz curve associated with the ú-exponential distribution of type II is given by:
The Gini coefficient is<blockquote></blockquote>
The ú-exponential distribution of type II behaves asymptotically as follows:
The ú-exponential distribution has been applied in several areas, such as: