shannon (version 0.2.0)

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

Description

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

Usage

Se_gum(alpha, beta)
re_gum(alpha, beta, delta)
hce_gum(alpha, beta, delta)
ae_gum(alpha, beta, delta)

Value

The functions Se_gum, re_gum, hce_gum, and ae_gum provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Gumbel distribution and \(\delta\).

Arguments

alpha

The location parameter of the Gumbel distribution (\(\alpha\in\left(-\infty,+\infty\right)\)).

beta

The strictly positive scale parameter of the Gumbel distribution (\(\beta > 0\)).

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 <[e-mail]>, Christophe Chesneau <[e-mail]> and Farrukh Jamal .

Details

The following is the probability density function of the Gumbel distribution: $$f(x)=\frac{1}{\beta}e^{-(z+e^{-z})},$$ where \(z=\frac{x-\alpha}{\beta}\), \(x\in\left(-\infty,+\infty\right)\), \(\alpha\in\left(-\infty,+\infty\right)\) and \(\beta > 0\).

References

Gomez, Y. M., Bolfarine, H., & Gomez, H. W. (2019). Gumbel distribution with heavy tails and applications to environmental data. Mathematics and Computers in Simulation, 157, 115-129.

See Also

re_norm

Examples

Se_gum(1.2, 1.4)
delta <- c(2, 3)
re_gum(1.2, 0.4, delta)
hce_gum(1.2, 0.4, delta)
ae_gum(1.2, 0.4, delta)