Frechet distribution: Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Fréchet distribution

Description

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Fréchet distribution.

Usage

se_fre(alpha, beta, zeta)
re_fre(alpha, beta, zeta, delta)
hce_fre(alpha, beta, zeta, delta)
ae_fre(alpha, beta, zeta, delta)

Value

The functions se_fre, re_fre, hce_fre, and ae_fre provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Fréchet distribution distribution and $\delta$.

Arguments

alpha: The parameter of the Fréchet distribution ($\alpha>0$).
beta: The parameter of the Fréchet distribution ($\beta\in\left(-\infty,+\infty\right)$).
zeta: The parameter of the Fréchet distribution ($\zeta>0$).
delta: The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Author

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.

Details

The following is the probability density function of the Fréchet distribution:

$$ f(x)=\frac{\alpha}{\zeta}\left(\frac{x-\beta}{\zeta}\right)^{-1-\alpha}e^{-(\frac{x-\beta}{\zeta})^{-\alpha},} $$ where $x>\beta$, $\alpha>0$, $\zeta>0$ and $\beta\in\left(-\infty,+\infty\right)$. The Fréchet distribution is also known as inverse Weibull distribution and special case of the generalized extreme value distribution.

References

Abbas, K., & Tang, Y. (2015). Analysis of Fréchet distribution using reference priors. Communications in Statistics-Theory and Methods, 44(14), 2945-2956.

Examples

Run this code

se_fre(0.2, 1.4, 1.2)
delta <- c(2, 3)
re_fre(1.2, 0.4, 1.2, delta)
hce_fre(1.2, 0.4, 1.2, delta)
ae_fre(1.2, 0.4, 1.2, delta)

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