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

Description

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

Usage

se_kumnorm(mu, sigma, a, b)
re_kumnorm(mu, sigma, a, b, delta)
hce_kumnorm(mu, sigma, a, b, delta)
ae_kumnorm(mu, sigma, a, b, delta)

Value

The functions se_kumnorm, re_kumnorm, hce_kumnorm, and ae_kumnorm provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Kumaraswamy normal distribution and $\delta$.

Arguments

mu: The location parameter of the normal distribution ($\mu\in\left(-\infty,+\infty\right)). $
sigma: The strictly positive scale parameter of the normal distribution ($\sigma > 0$).
a: The strictly positive shape parameter of the Kumaraswamy distribution ($a > 0$).
b: The strictly positive shape parameter of the Kumaraswamy distribution ($b > 0$).
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 Kumaraswamy normal distribution: $$ f(x)=\frac{ab}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)\left[\Phi\left(\frac{x-\mu}{\sigma}\right)\right]^{a-1}\left[1-\Phi\left(\frac{x-\mu}{\sigma}\right)^{a}\right]^{b-1}, $$ where $x\in\left(-\infty,+\infty\right)$, $\mu\in\left(-\infty,+\infty\right)$, $\sigma > 0$, $a > 0$ and $b > 0$, and the functions $\phi(t)$ and $\Phi(t) $, denote the probability density function and cumulative distribution function of the standard normal distribution, respectively.

References

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898.

Examples

Run this code

se_kumnorm(0.2, 1.5, 1, 1)
delta <- c(1.5, 2, 3)
re_kumnorm(1.2, 1, 2, 1.5, delta)
hce_kumnorm(1.2, 1, 2, 1.5, delta)
ae_kumnorm(1.2, 1, 2, 1.5, delta)

Run the code above in your browser using DataLab