Package 'shannon'

Computation of Entropy Measures and Relative Loss

Description

The functions allow for the numerical evaluation of some commonly used entropy measures, such as Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, at selected parametric values from several well-known and widely used probability distributions. Moreover, the functions also compute the relative loss of these entropies using the truncated distributions. Let $X$ be an absolutely continuous random variable having the probability density function $f(x)$. Then, the Shahnon entropy is as follows:

$H(X)=-\intop_{-\infty}^{+\infty}f(x)\log f(x)dx.$

The Rényi entropy is as follows:

$H_{\delta}(X)=\frac{1}{1-\delta}\log\intop_{-\infty}^{+\infty}f(x)^{\delta}dx;\qquad\delta>0,\delta\ne1.$

The Havrda and Charvat entropy is as follows:

$H_{\delta}(X)=\frac{1}{2^{1-\delta}-1}\left(\intop_{-\infty}^{+\infty}f(x)^{\delta}dx-1\right);\qquad\delta>0,\delta\ne1.$

The Arimoto entropy is as follows:

$H_{\delta}(X)=\frac{\delta}{1-\delta}\left[\left(\intop_{-\infty}^{+\infty}f(x)^{\delta}dx\right)^{\frac{1}{\delta}}-1\right];\qquad\delta>0,\delta\ne1.$

Let $D(X)$ be an entropy, and $D_p(X)$ be its truncated integral version at $p$, i.e., defined with the truncated version of $f(x)$ over the interval $(-\infty,p)$. Then we define the corresponding relative loss entropy is defined by

$S_D(p)= \frac{D(X)-D_p(X)}{D(X)}.$

Details

Package:	shannon
Type:	Package
Version:	0.2.0
Date:	2024-08-21
License:	GPL-2

Maintainers

Muhammad Imran <[email protected]>

Author(s)

Muhammad Imran [email protected], Christophe Chesneau [email protected] and Farrukh Jamal [email protected].

References

Shannon, C. E. (1948). A mathematical theory of communication. The Bell system technical journal, 27(3), 379-423.

Rényi, A. (1961). On measures of entropy and information, Hungarian Academy of Sciences, Budapest, Hungary, 547- 561.

Havrda, J., & Charvat, F. (1967). Quantification method of classification processes. Concept of structural $\alpha$-entropy. Kybernetika, 3(1), 30-35.

Arimoto, S. (1971). Information-theoretical considerations on estimation problems. Information and control, 19(3), 181-194.

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148.

---

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

Description

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

Usage

Se_beta(alpha, beta)
re_beta(alpha, beta, delta)
hce_beta(alpha, beta, delta)
ae_beta(alpha, beta, delta)

Arguments

alpha	The strictly positive shape parameter of the beta distribution ($\alpha > 0$).
beta	The strictly positive shape parameter of the beta distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=\frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha\right)\Gamma\left(\beta\right)}x^{\alpha-1}\left(1-x\right)^{\beta-1},$

where $0\leq x\leq1$, $\alpha > 0$ and $\beta > 0$, and $\Gamma(a)$ denotes the standard gamma function.

Value

The functions Se_beta, re_beta, hce_beta, and ae_beta provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the beta distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Gupta, A. K., & Nadarajah, S. (2004). Handbook of beta distribution and its applications. CRC Press.

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). Beta distributions. Continuous univariate distributions. 2nd ed. New York, NY: John Wiley and Sons, 221-235.

See Also

se_kum, re_kum, hce_kum, ae_kum

Examples

# Computation of the Shannon entropy
Se_beta(2, 4)
delta <- c(1.2, 3)
# Computation of the Rényi  entropy
re_beta(2, 4, delta)
# Computation of the Havrda and Charvat entropy
hce_beta(2, 4, delta)
# Computation of the Arimoto entropy
ae_beta(2, 4, delta)
# A graphic presentation of the Havrda and Charvat entropy (HCE)
library(ggplot2)
delta <- c(0.2, 0.3, 0.5, 0.8, 1.2, 1.5, 2.5, 3, 3.5)
hce_beta(2, 1.2, delta)
z <- hce_beta(2, 1.2, delta)
dat <- data.frame(x = delta , HCE = z)
p_hce <- ggplot(dat, aes(x = delta, y = HCE)) + geom_line()
plot <- p_hce + ggtitle(expression(alpha == 2~~beta == 1.2))

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the beta exponential distribution

Description

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

Usage

se_bexp(lambda, alpha, beta)
re_bexp(lambda, alpha, beta, delta)
hce_bexp(lambda, alpha, beta, delta)
ae_bexp(lambda, alpha, beta, delta)

Arguments

lambda	The strictly positive scale parameter of the exponential distribution ($\lambda > 0$).
alpha	The strictly positive shape parameter of the beta distribution ($\alpha > 0$).
beta	The strictly positive shape parameter of the beta distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the beta exponential distribution:

$f(x)=\frac{\lambda e^{-\beta\lambda x}}{B(\alpha,\beta)}\left(1-e^{-\lambda x}\right)^{\alpha-1},$

where $x > 0$, $\alpha > 0$, $\beta > 0$ and $\lambda > 0$, and $B(a,b)$ denotes the standard beta function.

Value

The functions se_bexp, re_bexp, hce_bexp, and ae_bexp provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the beta exponential distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Nadarajah, S., & Kotz, S. (2006). The beta exponential distribution. Reliability Engineering & System Safety, 91(6), 689-697.

See Also

re_beta, re_exp

Examples

# Computation of the Shannon entropy
se_bexp(1.2, 0.2, 1.5)
delta <- c(0.2, 0.3, 0.5)
# Computation of the Rényi entropy
re_bexp(1.2, 0.2, 0.5, delta)
# Computation of the Havrda and Charvat entropy
hce_bexp(1.2, 0.2, 1.5, delta)
# Computation of the Arimoto entropy
ae_bexp(1.2, 0.2, 1.5, delta)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Birnbaum-Saunders distribution

Description

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Birnbaum-Saunders distribution.

Usage

se_bs(v)
re_bs(v, delta)
hce_bs(v, delta)
ae_bs(v, delta)

Arguments

v	The strictly positive scale parameter of the Birnbaum-Saunders distribution ($v > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the Birnbaum-Saunders distribution:

$f(x)=\frac{x^{0.5}+x^{-0.5}}{2vx}\phi\left(\frac{x^{0.5}-x^{-0.5}}{v}\right),$

where $x > 0$ and $v > 0$, and $\phi(x)$ is the probability density function of the standard normal distribution.

Value

The functions se_bs, re_bs, hce_bs, and ae_bs provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Birnbaum-Saunders distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Chan, S., Nadarajah, S., & Afuecheta, E. (2016). An R package for value at risk and expected shortfall. Communications in Statistics Simulation and Computation, 45(9), 3416-3434.

Arimoto, S. (1971). Information-theoretical considerations on estimation problems. Inf. Control, 19, 181–194.

See Also

re_exp, re_chi

Examples

se_bs(0.2)
delta <- c(1.5, 2, 3)
re_bs(0.2, delta)
hce_bs(0.2, delta)
ae_bs(0.2, delta)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Burr XII distribution

Description

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

Usage

se_burr(k, c)
re_burr(k, c, delta)
hce_burr(k, c, delta)
ae_burr(k, c, delta)

Arguments

k	The strictly positive shape parameter of the Burr XII distribution ($k > 0$).
c	The strictly positive shape parameter of the Burr XII distribution ($c > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the Burr XII distribution:

$f(x)=kcx^{c-1}\left(1+x^{c}\right)^{-k-1},$

where $x > 0$, $c > 0$ and $k > 0$.

Value

The functions se_burr, re_burr, hce_burr, and ae_burr provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Burr XII distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Rodriguez, R. N. (1977). A guide to the Burr type XII distributions. Biometrika, 64(1), 129-134.

Zimmer, W. J., Keats, J. B., & Wang, F. K. (1998). The Burr XII distribution in reliability analysis. Journal of Quality Technology, 30(4), 386-394.

See Also

re_gamma, re_wei

Examples

se_burr(0.2, 1.4)
delta <- c(2, 3)
re_burr(1.2, 1.4, delta)
hce_burr(1.2, 1.4, delta)
ae_burr(1.2, 1.4, delta)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Chi-squared distribution

Description

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the chi-squared distribution.

Usage

se_chi(n)
re_chi(n, delta)
hce_chi(n, delta)
ae_chi(n, delta)

Arguments

n	The degree of freedom and the positive parameter of the Chi-squared distribution ($n > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the (non-central) Chi-squared distribution:

$f(x)=\frac{1}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}x^{\frac{n}{2}-1}e^{-\frac{x}{2}},$

where $x > 0$ and $n > 0$, and $\Gamma(a)$ denotes the standard gamma function.

Value

The functions se_chi, re_chi, hce_chi, and ae_chi provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Chi-squared distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John Wiley & Sons.

See Also

re_exp, re_gamma, re_bs

Examples

se_chi(1.2)
delta <- c(0.2, 0.3)
re_chi(1.2, delta)
hce_chi(1.2, delta)
ae_chi(1.2, delta)

---

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

Description

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

Usage

Se_exp(alpha)
re_exp(alpha, delta)
hce_exp(alpha, delta)
ae_exp(alpha, delta)

Arguments

alpha	The strictly positive scale parameter of the exponential distribution ($\alpha > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=\alpha e^{-\alpha x},$

where $x > 0$ and $\alpha > 0$.

Value

The functions Se_exp, re_exp, hce_exp, and ae_exp provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the exponential distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Balakrishnan, K. (2019). Exponential distribution: theory, methods and applications. Routledge.

Singh, A. K. (1997). The exponential distribution-theory, methods and applications, Technometrics, 39(3), 341-341.

Arimoto, S. (1971). Information-theoretical considerations on estimation problems. Inf. Control, 19, 181–194.

See Also

re_chi, re_gamma, re_wei

Examples

Se_exp(0.2)
delta <- c(1.5, 2, 3)
re_exp(0.2, delta)
hce_exp(0.2, delta)
ae_exp(0.2, delta)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the exponential extension distribution

Description

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

Usage

se_nh(alpha, beta)
re_nh(alpha, beta, delta)
hce_nh(alpha, beta, delta)
ae_nh(alpha, beta, delta)

Arguments

alpha	The strictly positive parameter of the exponential extension distribution ($\alpha > 0$).
beta	The strictly positive parameter of the exponential extension distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the exponential extension distribution:

$f(x)=\alpha\beta(1+\alpha x)^{\beta-1}e^{1-(1+\alpha x)^{\beta}},$

where $x > 0$, $\alpha > 0$ and $\beta > 0$.

Value

The functions se_nh, re_nh, hce_nh, and ae_nh provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the exponential extension distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Nadarajah, S., & Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6), 543-558.

See Also

re_exp, re_gamma, re_ee, re_wei

Examples

se_nh(1.2, 0.2)
delta <- c(1.5, 2, 3)
re_nh(1.2, 0.2, delta)
hce_nh(1.2, 0.2, delta)
ae_nh(1.2, 0.2, delta)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the exponentiated exponential distribution

Description

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

Usage

se_ee(alpha, beta)
re_ee(alpha, beta, delta)
hce_ee(alpha, beta, delta)
ae_ee(alpha, beta, delta)

Arguments

alpha	The strictly positive scale parameter of the exponentiated exponential distribution ($\alpha > 0$).
beta	The strictly positive shape parameter of the exponentiated exponential distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the exponentiated exponential distribution:

$f(x)=\alpha\beta e^{-\alpha x}\left(1-e^{-\alpha x}\right)^{\beta-1},$

where $x > 0$, $\alpha > 0$ and $\beta > 0$.

Value

The functions se_ee, re_ee, hce_ee, and ae_ee provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the exponentiated exponential distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Nadarajah, S. (2011). The exponentiated exponential distribution: a survey. AStA Advances in Statistical Analysis, 95, 219-251.

Gupta, R. D., & Kundu, D. (2007). Generalized exponential distribution: Existing results and some recent developments. Journal of Statistical Planning and Inference, 137(11), 3537-3547.

See Also

re_exp, re_wei, re_nh

Examples

se_ee(0.2, 1.4)
delta <- c(1.5, 2, 3)
re_ee(0.2, 1.4, delta)
hce_ee(0.2, 1.4, delta)
ae_ee(0.2, 1.4, delta)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the exponentiated Weibull distribution

Description

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

Usage

se_ew(a, beta, zeta)
re_ew(a, beta, zeta, delta)
hce_ew(a, beta, zeta, delta)
ae_ew(a, beta, zeta, delta)

Arguments

a	The strictly positive shape parameter of the exponentiated Weibull distribution ($a > 0$).
beta	The strictly positive scale parameter of the baseline Weibull distribution ($\beta > 0$).
zeta	The strictly positive shape parameter of the baseline Weibull distribution ($\zeta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the exponentiated Weibull distribution:

$f(x)=a\zeta\beta^{-\zeta}x^{\zeta-1}e^{-\left(\frac{x}{\beta}\right)^{\zeta}}\left[1-e^{-\left(\frac{x}{\beta}\right)^{\zeta}}\right]^{a-1},$

where $x > 0$, $a > 0$, $\beta > 0$ and $\zeta > 0$.

Value

The functions se_ew, re_ew, hce_ew, and ae_ew provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the exponentiated Weibull distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2013). The exponentiated Weibull distribution: a survey. Statistical Papers, 54, 839-877.

See Also

re_exp, re_wei, re_ew

Examples

se_ew(0.8, 0.2, 0.8)
delta <- c(1.5, 2, 3)
re_ew(1.2, 1.2, 1.4, delta)
hce_ew(1.2, 1.2, 1.4, delta)
ae_ew(1.2, 1.2, 1.4, delta)

---

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

Description

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

Usage

se_f(alpha, beta)
re_f(alpha, beta, delta)
hce_f(alpha, beta, delta)
ae_f(alpha, beta, delta)

Arguments

alpha	The strictly positive parameter (first degree of freedom) of the F distribution ($\alpha > 0$).
beta	The strictly positive parameter (second degree of freedom) of the F distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=\frac{1}{B(\frac{\alpha}{2},\frac{\beta}{2})}\left(\frac{\alpha}{\beta}\right)^{\frac{\alpha}{2}}x^{\frac{\alpha}{2}-1}\left(1+\frac{\alpha}{\beta}x\right)^{-\left(\frac{\alpha+\beta}{2}\right)},$

where $x > 0$, $\alpha > 0$ and $\beta > 0$, and $B(a,b)$ is the standard beta function.

Value

The functions se_f, re_f, hce_f, and ae_f provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the F distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John Wiley & Sons.

See Also

re_exp, re_gamma

Examples

se_f(1.2, 1.4)
delta <- c(2.2, 2.3)
re_f(1.2, 0.4, delta)
hce_f(1.2, 1.4, delta)
ae_f(1.2, 1.4, delta)

---

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)

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 ($\delta \ne 1$).

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.

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 $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

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

See Also

re_exp, re_gum

Examples

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)

---

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

Description

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

Usage

Se_gamma(alpha, beta)
re_gamma(alpha, beta, delta)
hce_gamma(alpha, beta, delta)
ae_gamma(alpha, beta, delta)

Arguments

alpha	The strictly positive shape parameter of the gamma distribution ($\alpha > 0$).
beta	The strictly positive scale parameter of the gamma distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x},$

where $x > 0$, $\alpha > 0$ and $\beta > 0$, and $\Gamma(a)$ is the standard gamma function.

Value

The functions Se_gamma, re_gamma, hce_gamma, and ae_gamma provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the gamma distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Burgin, T. A. (1975). The gamma distribution and inventory control. Journal of the Operational Research Society, 26(3), 507-525.

See Also

re_exp, re_wei

Examples

Se_gamma(1.2, 1.4)
delta <- c(1.5, 2, 3)
re_gamma(1.2, 1.4, delta)
hce_gamma(1.2, 1.4, delta)
ae_gamma(1.2, 1.4, delta)

---

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

Description

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

Usage

se_gomp(alpha, beta)
re_gomp(alpha, beta, delta)
hce_gomp(alpha, beta, delta)
ae_gomp(alpha, beta, delta)

Arguments

alpha	The strictly positive parameter of the Gompertz distribution ($\alpha > 0$).
beta	The strictly positive parameter of the Gompertz distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=\alpha e^{\beta x-\frac{\alpha}{\beta}\left(e^{\beta x}-1\right)},$

where $x > 0$, $\alpha > 0$ and $\beta > 0$.

Value

The functions se_gomp, re_gomp, hce_gomp, and ae_gomp provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Gompertz distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Soliman, A. A., Abd-Ellah, A. H., Abou-Elheggag, N. A., & Abd-Elmougod, G. A. (2012). Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data. Computational Statistics & Data Analysis, 56(8), 2471-2485.

See Also

re_exp, re_gamma, re_ray

Examples

se_gomp(2.4,0.2)
delta <- c(2, 3)
re_gomp(2.4,0.2, delta)
hce_gomp(2.4,0.2, delta)
ae_gomp(2.4,0.2, delta)

---

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)

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 ($\delta \ne 1$).

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 $\beta > 0$.

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 $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

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)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the inverse-gamma distribution

Description

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the inverse-gamma distribution.

Usage

se_ig(alpha, beta)
re_ig(alpha, beta, delta)
hce_ig(alpha, beta, delta)
ae_ig(alpha, beta, delta)

Arguments

alpha	The strictly positive shape parameter of the inverse-gamma distribution ($\alpha > 0$).
beta	The strictly positive scale parameter of the inverse-gamma distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the inverse-gamma distribution:

$f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{-\alpha-1}e^{-\frac{\beta}{x}},$

where $x > 0$, $\alpha > 0$ and $\beta > 0$, and $\Gamma(a)$ is the standard gamma function.

Value

The functions se_ig, re_ig, hce_ig, and ae_ig provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the inverse-gamma distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Rivera, P. A., Calderín-Ojeda, E., Gallardo, D. I., & Gómez, H. W. (2021). A compound class of the inverse Gamma and power series distributions. Symmetry, 13(8), 1328.

Glen, A. G. (2017). On the inverse gamma as a survival distribution. Computational Probability Applications, 15-30.

See Also

re_exp, re_gamma

Examples

se_ig(1.2, 0.2)
delta <- c(1.5, 2, 3)
re_ig(1.2, 0.2, delta)
hce_ig(1.2, 0.2, delta)
ae_ig(1.2, 0.2, delta)

---

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

Description

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

Usage

se_kum(alpha, beta)
re_kum(alpha, beta, delta)
hce_kum(alpha, beta, delta)
ae_kum(alpha, beta, delta)

Arguments

alpha	The strictly positive shape parameter of the Kumaraswamy distribution ($\alpha > 0$).
beta	The strictly positive shape parameter of the Kumaraswamy distribution ($\beta > 0$).
delta	The strictly positive scale parameter ($\delta > 0$).

Details

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

$f(x)=\alpha\beta x^{\alpha-1}\left(1-x^{\alpha}\right)^{\beta-1},$

where $0\leq x\leq1$, $\alpha > 0$ and $\beta > 0$.

Value

The functions se_kum, re_kum, hce_kum, and ae_kum provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Kumaraswamy distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

El-Sherpieny, E. S. A., & Ahmed, M. A. (2014). On the Kumaraswamy distribution. International Journal of Basic and Applied Sciences, 3(4), 372.

Al-Babtain, A. A., Elbatal, I., Chesneau, C., & Elgarhy, M. (2021). Estimation of different types of entropies for the Kumaraswamy distribution. PLoS One, 16(3), e0249027.

See Also

re_beta

Examples

se_kum(1.2, 1.4)
delta <- c(1.5, 2, 3)
re_kum(1.2, 1.4, delta)
hce_kum(1.2, 1.4, delta)
ae_kum(1.2, 1.4, delta)

---

Compute the Rényi, Havrda and Charvat, and Arimoto entropies of the Kumaraswamy exponential distribution

Description

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

Usage

re_kexp(lambda, a, b, delta)
hce_kexp(lambda, a, b, delta)
ae_kexp(lambda, a, b, delta)

Arguments

a	The strictly positive shape parameter of the Kumaraswamy distribution ($a > 0$).
b	The strictly positive shape parameter of the Kumaraswamy distribution ($b > 0$).
lambda	The strictly positive parameter of the exponential distribution ($\lambda > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the Kumaraswamy exponential distribution:

$f(x)=ab\lambda e^{-\lambda x}\left(1-e^{-\lambda x}\right)^{a-1}\left\{ 1-\left(1-e^{-\lambda x}\right)^{a}\right\} ^{b-1},$

where $x > 0$, $a > 0$, $b > 0$ and $\lambda > 0$.

Value

The functions re_kexp, hce_kexp, and ae_kexp provide the Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Kumaraswamy exponential distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

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

See Also

re_exp, re_kum

Examples

delta <- c(1.5, 2, 3)
re_kexp(1.2, 1.2, 1.4, delta)
hce_kexp(1.2, 1.2, 1.4, delta)
ae_kexp(1.2, 1.2, 1.4, delta)

---

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)

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 ($\delta \ne 1$).

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.

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 $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

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

See Also

re_norm, re_kum

Examples

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)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Laplace or the double exponential distributiondistribution

Description

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

Usage

Se_lap(alpha, beta)
re_lap(alpha, beta, delta)
hce_lap(alpha, beta, delta)
ae_lap(alpha, beta, delta)

Arguments

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

Details

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

$f(x)=\frac{1}{2\beta}e^{\frac{-|x-\alpha|}{\beta}},$

where $x\in\left(-\infty,+\infty\right)$, $\alpha\in\left(-\infty,+\infty\right)$ and $\beta > 0$.

Value

The functions Se_lap, re_lap, hce_lap, and ae_lap provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Laplace distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Cordeiro, G. M., & Lemonte, A. J. (2011). The beta Laplace distribution. Statistics & Probability Letters, 81(8), 973-982.

See Also

re_gum, re_norm

Examples

Se_lap(0.2, 1.4)
delta <- c(2, 3)
re_lap(1.2, 0.4, delta)
hce_lap(1.2, 0.4, delta)
ae_lap(1.2, 0.4, delta)

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the log-normal distribution

Description

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

Usage

se_lnorm(mu, sigma)
re_lnorm(mu, sigma, delta)
hce_lnorm(mu, sigma, delta)
ae_lnorm(mu, sigma, delta)

Arguments

mu	The location parameter ($\mu\in\left(-\infty,+\infty\right)$).
sigma	The strictly positive scale parameter of the log-normal distribution ($\sigma > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the log-normal distribution:

$f(x)=\frac{1}{x\sigma\sqrt{2\pi}}e^{-\frac{\left(\log(x)-\mu\right)^{2}}{2\sigma^{2}}},$

where $x > 0$, $\mu\in\left(-\infty,+\infty\right)$ and $\sigma > 0$.

Value

The functions se_lnorm, re_lnorm, hce_lnorm, and ae_lnorm provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the log-normal distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, Volume 1, Chapter 14. Wiley, New York.

See Also

re_wei, re_norm

Examples

se_lnorm(0.2, 1.4)
delta <- c(2, 3)
re_lnorm(1.2, 0.4, delta)
hce_lnorm(1.2, 0.4, delta)
ae_lnorm(1.2, 0.4, delta)

---

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

Description

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

Usage

se_logis(mu, sigma)
re_logis(mu, sigma, delta)
hce_logis(mu, sigma, delta)
ae_logis(mu, sigma, delta)

Arguments

mu	The location parameter of the logistic distribution ($\mu\in\left(-\infty,+\infty\right)$).
sigma	The strictly positive scale parameter of the logistic distribution ($\sigma > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=\frac{e^{-\frac{\left(x-\mu\right)}{\sigma}}}{\sigma\left(1+e^{-\frac{\left(x-\mu\right)}{\sigma}}\right)^{2}},$

where $x\in\left(-\infty,+\infty\right)$, $\mu\in\left(-\infty,+\infty\right)$ and $\sigma > 0$.

Value

The functions se_logis, re_logis, hce_logis, and ae_logis provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the logistic distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, Volume 2 (Vol. 289). John Wiley & Sons.

See Also

re_gum, re_norm

Examples

se_logis(0.2, 1.4)
delta <- c(2, 3)
re_logis(1.2, 0.4, delta)
hce_logis(1.2, 0.4, delta)
ae_logis(1.2, 0.4, delta)

---

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

Description

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

Usage

se_lom(alpha, beta)
re_lom(alpha, beta, delta)
hce_lom(alpha, beta, delta)
ae_lom(alpha, beta, delta)

Arguments

alpha	The strictly positive shape parameter of the Lomax distribution ($\alpha > 0$).
beta	The strictly positive scale parameter of the Lomax distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=\frac{\alpha}{\beta}\left(1+\frac{x}{\beta}\right)^{-\alpha-1},$

where $x > 0$, $\alpha > 0$ and $\beta > 0$.

Value

The functions se_lom, re_lom, hce_lom, and ae_lom provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Lomax distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Abd-Elfattah, A. M., Alaboud, F. M., & Alharby, A. H. (2007). On sample size estimation for Lomax distribution. Australian Journal of Basic and Applied Sciences, 1(4), 373-378.

See Also

re_exp, re_gamma

Examples

se_lom(1.2, 0.2)
delta <- c(1.5, 2, 3)
re_lom(1.2, 0.2, delta)
hce_lom(1.2, 0.2, delta)
ae_lom(1.2, 0.2, delta)

---

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

Description

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

Usage

se_naka(alpha, beta)
re_naka(alpha, beta, delta)
hce_naka(alpha, beta, delta)
ae_naka(alpha, beta, delta)

Arguments

alpha	The strictly positive scale parameter of the Nakagami distribution ($\alpha > 0$).
beta	The strictly positive shape parameter of the Nakagami distribution ($\beta > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=\frac{2\alpha^{\alpha}}{\Gamma(\alpha)\beta^{\alpha}}x^{2\alpha-1}e^{-\frac{\alpha x^{2}}{\beta}},$

where $x > 0$, $\alpha > 0$ and $\beta > 0$, and $\Gamma(a)$ is the standard gamma function.

Value

The functions se_naka, re_naka, hce_naka, and ae_naka provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Nakagami distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Schwartz, J., Godwin, R. T., & Giles, D. E. (2013). Improved maximum-likelihood estimation of the shape parameter in the Nakagami distribution. Journal of Statistical Computation and Simulation, 83(3), 434-445.

See Also

re_exp, re_gamma, re_wei

Examples

se_naka(1.2, 0.2)
delta <- c(1.5, 2, 3)
re_naka(1.2, 0.2, delta)
hce_naka(1.2, 0.2, delta)
ae_naka(1.2, 0.2, delta)

---

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

Description

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

Usage

se_norm(alpha, beta)
re_norm(alpha, beta, delta)
hce_norm(alpha, beta, delta)
ae_norm(alpha, beta, delta)

Arguments

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

Details

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

$f(x)=\frac{1}{\beta\sqrt{2\pi}}e^{-0.5\left(\frac{x-\alpha}{\beta}\right)^{2}},$

where $x\in\left(-\infty,+\infty\right)$, $\alpha\in\left(-\infty,+\infty\right)$ and $\beta > 0$. The parameters $\alpha$ and $\beta$ represent the mean and standard deviation, respectively.

Value

The functions se_norm, re_norm, hce_norm, and ae_norm provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Normal distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Patel, J. K., & Read, C. B. (1996). Handbook of the normal distribution (Vol. 150). CRC Press.

See Also

re_gum

Examples

se_norm(0.2, 1.4)
delta <- c(1.5, 2, 3)
re_norm(0.2, 1.4, delta)
hce_norm(0.2, 1.4, delta)
ae_norm(0.2, 1.4, delta)

---

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

Description

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

Usage

se_ray(alpha)
re_ray(alpha, delta)
hce_ray(alpha, delta)
ae_ray(alpha, delta)

Arguments

alpha	The strictly positive parameter of the Rayleigh distribution ($\alpha > 0$).
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

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

$f(x)=2\alpha xe^{-\alpha x^{2}},$

where $x > 0$ and $\alpha > 0$.

Value

The functions se_ray, re_ray, hce_ray, and ae_ray provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Rayleigh distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Dey, S., Maiti, S. S., & Ahmad, M. (2016). Comparison of different entropy measures. Pak. J. Statist, 32(2), 97-108.

Arimoto, S. (1971). Information-theoretical considerations on estimation problems. Inf. Control, 19, 181–194.

See Also

re_exp, re_gamma, re_wei

Examples

se_ray(0.2)
delta <- c(1.5, 2, 3)
re_ray(0.2, delta)
hce_ray(0.2, delta)
ae_ray(0.2, delta)
# A graphic representation of the Rényi entropy (RE)
library(ggplot2)
delta <- c(1.5, 2, 3)
z <- re_ray(0.2, delta)
dat <- data.frame(x = delta , RE = z)
p_re <- ggplot(dat, aes(x = delta, y = RE)) + geom_line()
plot <- p_re + ggtitle(expression(alpha == 0.2))
# A graphic presentation of the Havrda and Charvat entropy (HCE)
delta <- c(1.5, 2, 3)
z <- hce_ray(0.2, delta)
dat <- data.frame(x = delta , HCE = z)
p_hce <- ggplot(dat, aes(x = delta, y = HCE)) + geom_line()
plot <- p_hce + ggtitle(expression(alpha == 0.2))

---

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Student's t distribution

Description

Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Student's t distribution.

Usage

se_st(v)
re_st(v, delta)
hce_st(v, delta)
ae_st(v, delta)

Arguments

v	The strictly positive parameter of the Student's t distribution ($v > 0$), also called a degree of freedom.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Details

The following is the probability density function of the Student t distribution:

$f(x)=\frac{\Gamma(\frac{v+1}{2})}{\sqrt{v\pi}\Gamma(\frac{v}{2})}\left(1+\frac{x^{2}}{v}\right)^{-(v+1)/2},$

where $x\in\left(-\infty,+\infty\right)$ and $v > 0$, and $\Gamma(a)$ is the standard gamma function.

Value

The functions se_st, re_st, hce_st, and ae_st provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Student's t distribution and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Yang, Z., Fang, K. T., & Kotz, S. (2007). On the Student's t-distribution and the t-statistic. Journal of Multivariate Analysis, 98(6), 1293-1304.

Ahsanullah, M., Kibria, B. G., & Shakil, M. (2014). Normal and Student's t distributions and their applications (Vol. 4). Paris, France: Atlantis Press.

Arimoto, S. (1971). Information-theoretical considerations on estimation problems. Inf. Control, 19, 181–194.

See Also

re_exp, re_gamma

Examples

se_st(4)
delta <- c(1.5, 2, 3)
re_st(4, delta)
hce_st(4, delta)
ae_st(4, delta)

---

Relative loss for various entropy measures using the truncated beta distribution

Description

Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated beta distribution.

Usage

rlse_beta(p, alpha, beta)
rlre_beta(p, alpha, beta, delta)
rlhce_beta(p, alpha, beta, delta)
rlae_beta(p, alpha, beta, delta)

Arguments

alpha	The strictly positive shape parameter of the beta distribution ($\alpha > 0$).
beta	The strictly positive shape parameter of the beta distribution ($\beta > 0$).
p	The truncation time $(p>0)$.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Value

The functions rlse_beta, rlre_beta, rlhce_beta, and rlae_beta provide the relative information loss based on the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the truncated beta distribution, $p$ and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Gupta, A. K., & Nadarajah, S. (2004). Handbook of beta distribution and its applications. CRC Press.

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148.

See Also

re_beta

Examples

p <- c(0.25, 0.50, 0.75)
rlse_beta(p, 0.2, 0.4)
rlre_beta(p, 0.2, 0.4, 0.5)
rlhce_beta(p, 0.2, 0.4, 0.5)
rlae_beta(p, 0.2, 0.4, 0.5)

---

Relative loss for various entropy measures using the truncated Birnbaum-Saunders distribution

Description

Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Birnbaum-Saunders distribution.

Usage

rlse_bs(p, v)
rlre_bs(p, v, delta)
rlhce_bs(p, v, delta)
rlae_bs(p, v, delta)

Arguments

v	The strictly positive scale parameter of the Birnbaum-Saunders distribution ($v > 0$).
p	The truncation time $(p>0)$.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Value

The functions rlse_bs, rlre_bs, rlhce_bs, and rlae_bs provide the relative information loss based on the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the truncated Birnbaum-Saunders distribution, $p$ and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148.

Chan, S., Nadarajah, S., & Afuecheta, E. (2016). An R package for value at risk and expected shortfall. Communications in Statistics Simulation and Computation, 45(9), 3416-3434.

Arimoto, S. (1971). Information-theoretical considerations on estimation problems. Inf. Control, 19, 181–194.

See Also

re_bs

Examples

p <- c(1, 1.7, 3)
rlse_bs(p, 0.2)
rlre_bs(p, 0.2, 0.5)
rlhce_bs(p, 0.2, 0.5)
rlae_bs(p, 0.2, 0.5)

---

Relative loss for various entropy measures using the truncated Chi-squared distribution

Description

Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Chi-squared distribution.

Usage

rlse_chi(p, n)
rlre_chi(p, n, delta)
rlhce_chi(p, n, delta)
rlae_chi(p, n, delta)

Arguments

n	The degree of freedom and positive parameter of the Chi-squared distribution ($n > 0$).
p	The truncation time $(p>0)$.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Value

The functions rlse_chi, rlre_chi, rlhce_chi, and rlae_chi provide the relative information loss based on the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the truncated Chi-squared distribution, $p$ and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148.

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John Wiley & Sons.

See Also

re_chi

Examples

p <- c(1, 1.7, 3)
rlse_chi(p, 2)
rlre_chi(p, 2, 0.5)
rlhce_chi(p, 2, 0.5)
rlae_chi(p, 2, 0.5)

---

Relative loss for various entropy measures using the truncated exponential distribution

Description

Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated exponential distribution.

Usage

rlse_exp(p, alpha)
rlre_exp(p, alpha, delta)
rlhce_exp(p, alpha, delta)
rlae_exp(p, alpha, delta)

Arguments

alpha	The strictly positive scale parameter of the exponential distribution ($\alpha > 0$).
p	The truncation time $(p>0)$.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Value

The functions rlse_exp, rlre_exp, rlhce_exp, and rlae_exp provide the relative information loss based on the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the truncated exponential distribution, $p$ and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148.

See Also

re_exp

Examples

p <- c(1, 1.7, 3)
rlse_exp(p, 2)
rlre_exp(p, 2, 0.5)
rlhce_exp(p, 2, 0.5)
rlae_exp(p, 2, 0.5)

---

Relative loss for various entropy measures using the truncated exponential extension distribution

Description

Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated exponential extension distribution.

Usage

rlse_nh(p, alpha, beta)
rlre_nh(p, alpha, beta, delta)
rlhce_nh(p, alpha, beta, delta)
rlae_nh(p, alpha, beta, delta)

Arguments

alpha	The strictly positive parameter of the exponential extension distribution ($\alpha > 0$).
beta	The strictly positive parameter of the exponential extension distribution ($\beta > 0$).
p	The truncation time $(p>0)$.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Value

The functions rlse_nh, rlre_nh, rlhce_nh, and rlae_nh provide the relative information loss based on the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the truncated exponential extension distribution, $p$ and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148.

Nadarajah, S., & Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6), 543-558.

See Also

re_nh

Examples

p <- c(0.25, 0.50, 0.75)
rlse_nh(p, 1.2, 0.2)
rlre_nh(p, 1.2, 0.2, 0.5)
rlhce_nh(p, 1.2, 0.2, 0.5)
rlae_nh(p, 1.2, 0.2, 0.5)

---

Relative loss for various entropy measures using the truncated F distribution

Description

Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated F distribution.

Usage

rlse_f(p, alpha, beta)
rlre_f(p, alpha, beta, delta)
rlhce_f(p, alpha, beta, delta)
rlae_f(p, alpha, beta, delta)

Arguments

alpha	The strictly positive parameter (first degree of freedom) of the F distribution ($\alpha > 0$).
beta	The strictly positive parameter (second degree of freedom) of the F distribution ($\beta > 0$).
p	The truncation time $(p>0)$.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Value

The functions rlse_f, rlre_f, rlhce_f, and rlae_f provide the relative information loss based on the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the truncated F distribution, $p$ and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148. Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John Wiley & Sons.

See Also

re_f

Examples

p <- c(1.25, 1.50, 1.75)
rlse_f(p, 4, 6)
rlre_f(p, 4, 6, 0.5)
rlhce_f(p, 4, 6, 0.5)
rlae_f(p, 4, 6, 0.5)

---

Relative loss for various entropy measures using the truncated gamma distribution

Description

Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated gamma distribution.

Usage

rlse_gamma(p, alpha, beta)
rlre_gamma(p, alpha, beta, delta)
rlhce_gamma(p, alpha, beta, delta)
rlae_gamma(p, alpha, beta, delta)

Arguments

alpha	The strictly positive shape parameter of the gamma distribution ($\alpha > 0$).
beta	The strictly positive scale parameter of the gamma distribution ($\beta > 0$).
p	The truncation time $(p>0)$.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Value

The functions rlse_gamma, rlre_gamma, rlhce_gamma, and rlae_gamma provide the relative information loss based on the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the truncated gamma distribution, $p$ and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148.

Burgin, T. A. (1975). The gamma distribution and inventory control. Journal of the Operational Research Society, 26(3), 507-525.

See Also

re_gamma

Examples

p <- c(1, 1.50, 1.75)
rlse_gamma(p, 0.2, 1)
rlre_gamma(p, 0.2, 1, 0.5)
rlhce_gamma(p, 0.2, 1, 0.5)
rlae_gamma(p, 0.2, 1, 0.5)

---

Relative loss for various entropy measures using the truncated Gompertz distribution

Description

Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Gompertz distribution.

Usage

rlse_gomp(p, alpha, beta)
rlre_gomp(p, alpha, beta, delta)
rlhce_gomp(p, alpha, beta, delta)
rlae_gomp(p, alpha, beta, delta)

Arguments

alpha	The strictly positive parameter of the Gompertz distribution ($\alpha > 0$).
beta	The strictly positive parameter of the Gompertz distribution ($\beta > 0$).
p	The truncation time $(p>0)$.
delta	The strictly positive parameter ($\delta > 0$) and ($\delta \ne 1$).

Value

The functions rlse_gomp, rlre_gomp, rlhce_gomp, and rlae_gomp provide the relative information loss based on the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the truncated Gompertz distribution, $p$ and $\delta$.

Author(s)

Muhammad Imran, Christophe Chesneau and Farrukh Jamal

R implementation and documentation: Muhammad Imran <[email protected]>, Christophe Chesneau <[email protected]> and Farrukh Jamal [email protected].

References

Awad, A. M., & Alawneh, A. J. (1987). Application of entropy to a life-time model. IMA Journal of Mathematical Control and Information, 4(2), 143-148.

Soliman, A. A., Abd-Ellah, A. H., Abou-Elheggag, N. A., & Abd-Elmougod, G. A. (2012). Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data. Computational Statistics & Data Analysis, 56(8), 2471-2485.

See Also

re_gomp

Examples

p <- c(0.25, 0.50)
rlse_gomp(p, 2.4,0.2)
rlre_gomp(p, 2.4,0.2, 0.5)
rlhce_gomp(p, 2.4,0.2, 0.5)
rlae_gomp(p, 2.4,0.2, 0.5)

---