| Type: | Package |
| Title: | Computation of Entropy Measures and Relative Loss |
| Version: | 0.2.0 |
| Author: | Muhammad Imran [aut, cre], Christophe Chesneau [aut], Farrukh Jamal [aut] |
| Maintainer: | Muhammad Imran <imranshakoor84@yahoo.com> |
| Depends: | R (≥ 4.0) |
| Imports: | stats, VaRES, extraDistr |
| Suggests: | ggplot2 |
| 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. Related works include: 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. <doi:10.1093/imamci/4.2.143>. |
| License: | GPL-2 |
| Encoding: | UTF-8 |
| RoxygenNote: | 7.2.3 |
| NeedsCompilation: | no |
| Packaged: | 2024-08-21 07:35:01 UTC; Amir Computers |
| Repository: | CRAN |
| Date/Publication: | 2024-08-26 16:00:02 UTC |
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 <imranshakoor84@yahoo.com>
Author(s)
Muhammad Imran imranshakoor84@yahoo.com, Christophe Chesneau christophe.chesneau@unicaen.fr and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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 ( |
beta |
The strictly positive shape parameter of the beta distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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 ( |
alpha |
The strictly positive shape parameter of the beta distribution ( |
beta |
The strictly positive shape parameter of the beta distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Nadarajah, S., & Kotz, S. (2006). The beta exponential distribution. Reliability Engineering & System Safety, 91(6), 689-697.
See Also
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 ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
c |
The strictly positive shape parameter of the Burr XII distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John Wiley & Sons.
See Also
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 ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive parameter of the exponential extension distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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 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 ( |
beta |
The strictly positive scale parameter of the baseline Weibull distribution ( |
zeta |
The strictly positive shape parameter of the baseline Weibull distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Nadarajah, S., Cordeiro, G. M., & Ortega, E. M. (2013). The exponentiated Weibull distribution: a survey. Statistical Papers, 54, 839-877.
See Also
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 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 ( |
beta |
The strictly positive shape parameter of the exponentiated exponential distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 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 ( |
beta |
The strictly positive parameter (second degree of freedom) of the F distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, volume 2 (Vol. 289). John Wiley & Sons.
See Also
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 ( |
beta |
The parameter of the Fréchet distribution ( |
zeta |
The parameter of the Fréchet distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive scale parameter of the gamma distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Burgin, T. A. (1975). The gamma distribution and inventory control. Journal of the Operational Research Society, 26(3), 507-525.
See Also
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 ( |
beta |
The strictly positive parameter of the Gompertz distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive scale parameter of the Gumbel distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive scale parameter of the inverse-gamma distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive shape parameter of the Kumaraswamy distribution ( |
delta |
The strictly positive scale parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
b |
The strictly positive shape parameter of the Kumaraswamy distribution ( |
lambda |
The strictly positive parameter of the exponential distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
sigma |
The strictly positive scale parameter of the normal distribution ( |
a |
The strictly positive shape parameter of the Kumaraswamy distribution ( |
b |
The strictly positive shape parameter of the Kumaraswamy distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive scale parameter of the Laplace distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Cordeiro, G. M., & Lemonte, A. J. (2011). The beta Laplace distribution. Statistics & Probability Letters, 81(8), 973-982.
See Also
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 ( |
sigma |
The strictly positive scale parameter of the log-normal distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, Volume 1, Chapter 14. Wiley, New York.
See Also
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 ( |
sigma |
The strictly positive scale parameter of the logistic distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions, Volume 2 (Vol. 289). John Wiley & Sons.
See Also
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 ( |
beta |
The strictly positive scale parameter of the Lomax distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive shape parameter of the Nakagami distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive scale parameter of the normal distribution ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
References
Patel, J. K., & Read, C. B. (1996). Handbook of the normal distribution (Vol. 150). CRC Press.
See Also
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 ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 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 ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 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 ( |
beta |
The strictly positive parameter (second degree of freedom) of the F distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 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 ( |
beta |
The strictly positive parameter of the Gompertz distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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)
Relative loss for various entropy measures using the truncated Gumbel distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Gumbel distribution.
Usage
rlse_gum(p, alpha, beta)
rlre_gum(p, alpha, beta, delta)
rlhce_gum(p, alpha, beta, delta)
rlae_gum(p, alpha, beta, delta)
Arguments
alpha |
The location parameter of the Gumbel distribution ( |
beta |
The strictly positive scale parameter of the Gumbel distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_gum, rlre_gum, rlhce_gum, and rlae_gum 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 Gumbel distribution, p and \delta.
Author(s)
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.
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.
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
Examples
p <- c(1.8,2.2)
rlse_gum(p, 4, 2)
rlre_gum(p, 4, 2, 2)
rlhce_gum(p, 4, 2, 2)
rlae_gum(p, 4, 2, 2)
Relative loss for various entropy measures using the truncated Kumaraswamy distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Kumaraswamy distribution.
Usage
rlse_kum(p, alpha, beta)
rlre_kum(p, alpha, beta, delta)
rlhce_kum(p, alpha, beta, delta)
rlae_kum(p, alpha, beta, delta)
Arguments
alpha |
The strictly positive shape parameter of the Kumaraswamy distribution ( |
beta |
The strictly positive shape parameter of the Kumaraswamy distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_kum, rlre_kum, rlhce_kum, and rlae_kum 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 Kumaraswamy distribution, p and \delta.
Author(s)
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.
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.
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
Examples
p <- c(0.25, 0.50, 0.75)
rlse_kum(p, 0.2, 0.4)
rlre_kum(p, 0.2, 0.4, 0.5)
rlhce_kum(p, 0.2, 0.4, 0.5)
rlae_kum(p, 0.2, 0.4, 0.5)
Relative loss for various entropy measures using the truncated Laplace distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Laplace distribution.
Usage
rlse_lap(p, alpha, beta)
rlre_lap(p, alpha, beta, delta)
rlhce_lap(p, alpha, beta, delta)
rlae_lap(p, alpha, beta, delta)
Arguments
alpha |
Location parameter of the Laplace distribution ( |
beta |
The strictly positive scale parameter of the Laplace distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_lap, rlre_lap, rlhce_lap, and rlae_lap 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 Laplace distribution, p and \delta.
Author(s)
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.
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.
Cordeiro, G. M., & Lemonte, A. J. (2011). The beta Laplace distribution. Statistics & Probability Letters, 81(8), 973-982.
See Also
Examples
p <- c(0.25, 0.50, 0.75)
rlse_lap(p, 0.2, 0.4)
rlre_lap(p, 0.2, 0.4, 0.5)
rlhce_lap(p, 0.2, 0.4, 0.5)
rlae_lap(p, 0.2, 0.4, 0.5)
Relative loss for various entropy measures using the truncated Nakagami distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Nakagami distribution.
Usage
rlse_naka(p, alpha, beta)
rlre_naka(p, alpha, beta, delta)
rlhce_naka(p, alpha, beta, delta)
rlae_naka(p, alpha, beta, delta)
Arguments
alpha |
The strictly positive scale parameter of the Nakagami distribution ( |
beta |
The strictly positive shape parameter of the Nakagami distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_naka, rlre_naka, rlhce_naka, and rlae_naka 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 Nakagami distribution, p and \delta.
Author(s)
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.
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.
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
Examples
p <- c(1.25, 1.50, 1.75)
rlse_naka(p, 0.2, 1)
rlre_naka(p, 0.2, 1, 0.5)
rlhce_naka(p, 0.2, 1, 0.5)
rlae_naka(p, 0.2, 1, 0.5)
Relative loss for various entropy measures using the truncated Rayleigh distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Rayleigh distribution.
Usage
rlse_ray(p, alpha)
rlre_ray(p, alpha, delta)
rlhce_ray(p, alpha, delta)
rlae_ray(p, alpha, delta)
Arguments
alpha |
The strictly positive scale parameter of the Rayleigh distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_ray, rlre_ray, rlhce_ray, and rlae_ray 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 Rayleigh distribution, p and \delta.
Author(s)
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.
References
Dey, S., Maiti, S. S., & Ahmad, M. (2016). Comparison of different entropy measures. Pak. J. Statist, 32(2), 97-108.
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
Examples
p <- seq(0.25, 2, by=0.25)
rlse_ray(p, 2)
rlre_ray(p, 2, 0.5)
rlhce_ray(p, 2, 0.5)
rlae_ray(p, 2, 0.5)
# A graphic representation of relative loss (RL)
library(ggplot2)
# p is a truncation time vector
p <- seq(0.25, 2, by = 0.25)
# RL based on the Rényi entropy
z1 <- rlre_ray(p, 0.1, 0.5)
# RL based on the Havrda and Charvat entropy
z2 <- rlhce_ray(p, 0.1, 0.5)
# RL based on the Arimoto entropy
z3 <- rlae_ray(p, 0.1, 0.5)
# RL based on the Shannon entropy
z4 <- rlse_ray(p, 0.1)
df <- data.frame(x = p, RL = z1, z2, z3, z4)
head(df)
p1 <- ggplot(df, aes(x = p, y = RL, color = Entropy))
p1 + geom_line(aes(colour = "RE"), size = 1) + geom_line(aes(x,
y = z2, colour = "HCE"), size = 1) + geom_line(aes(x, y = z3,
colour = "AR"), size = 1) + geom_line(aes(x, y = z4, colour = "SE"),
size = 1) + ggtitle(expression(delta == 0.5 ~ ~alpha == 0.1))
Relative loss for various entropy measures using the truncated Student's t distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Student's t distribution.
Usage
rlse_st(p, v)
rlre_st(p, v, delta)
rlhce_st(p, v, delta)
rlae_st(p, v, delta)
Arguments
v |
The strictly positive parameter of the Student distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_st, rlre_st, rlhce_st, and rlae_st 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 Student's t distribution, p and \delta.
Author(s)
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.
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.
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
Examples
p <- c(1, 1.7, 3)
rlse_st(p, 4)
rlre_st(p, 4, 0.5)
rlhce_st(p, 4, 0.5)
rlae_st(p, 4, 0.5)
Relative loss for various entropy measures using the truncated Weibull distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated Weibull distribution.
Usage
rlse_wei(p, alpha, beta)
rlre_wei(p, alpha, beta, delta)
rlhce_wei(p, alpha, beta, delta)
rlae_wei(p, alpha, beta, delta)
Arguments
alpha |
The strictly positive scale parameter of the Weibull distribution ( |
beta |
The strictly positive shape parameter of the Weibull distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_wei, rlre_wei, rlhce_wei, and rlae_wei 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 Weibull distribution, p and \delta.
Author(s)
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.
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.
Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of applied mechanics, 18, 293-297.
See Also
Examples
p <- c(1, 1.7, 3)
rlse_wei(p, 2, 1)
rlre_wei(p, 2, 1, 0.5)
rlhce_wei(p, 2, 1, 0.5)
rlae_wei(p, 2, 1, 0.5)
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 ( |
beta |
The strictly positive shape parameter of the beta distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 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 ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 ( |
beta |
The strictly positive parameter of the exponential extension distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 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 ( |
beta |
The strictly positive scale parameter of the gamma distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
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 <imranshakoor84@yahoo.com>, Christophe Chesneau <christophe.chesneau@unicaen.fr> and Farrukh Jamal farrukh.jamal@iub.edu.pk.
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
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 inverse-gamma distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated inverse-gamma distribution.
Usage
rlse_ig(p, alpha, beta)
rlre_ig(p, alpha, beta, delta)
rlhce_ig(p, alpha, beta, delta)
rlae_ig(p, alpha, beta, delta)
Arguments
alpha |
The strictly positive shape parameter of the inverse-gamma distribution ( |
beta |
The strictly positive scale parameter of the inverse-gamma distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_ig, rlre_ig, rlhce_ig, and rlae_ig 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 inverse-gamma distribution, p and \delta.
Author(s)
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.
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.
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.
See Also
Examples
p <- c(1.25, 1.50)
rlse_ig(p, 1.2, 0.2)
rlre_ig(p, 1.2, 0.2, 0.5)
rlhce_ig(p, 1.2, 0.2, 0.5)
rlae_ig(p, 1.2, 0.2, 0.5)
Relative loss for various entropy measures using the truncated normal distribution
Description
Compute the relative information loss of the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the truncated normal distribution.
Usage
rlse_norm(p, alpha, beta)
rlre_norm(p, alpha, beta, delta)
rlhce_norm(p, alpha, beta, delta)
rlae_norm(p, alpha, beta, delta)
Arguments
alpha |
The location parameter of the normal distribution ( |
beta |
The strictly positive scale parameter of the normal distribution ( |
p |
The truncation time |
delta |
The strictly positive parameter ( |
Value
The functions rlse_norm, rlre_norm, rlhce_norm, and rlae_norm 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 normal distribution, p and \delta.
Author(s)
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.
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.
Patel, J. K., & Read, C. B. (1996). Handbook of the normal distribution (Vol. 150). CRC Press.
See Also
Examples
p <- c(0.25, 0.50, 0.75)
rlse_norm(p, 0.2, 1)
rlre_norm(p, 0.2, 1, 0.5)
rlhce_norm(p, 0.2, 1, 0.5)
rlae_norm(p, 0.2, 1, 0.5)
Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Weibull distribution
Description
Compute the Shannon, Rényi, Havrda and Charvat, and Arimoto entropies of the Weibull distribution.
Usage
se_wei(alpha, beta)
re_wei(alpha, beta, delta)
hce_wei(alpha, beta, delta)
ae_wei(alpha, beta, delta)
Arguments
alpha |
The strictly positive scale parameter of the Weibull distribution ( |
beta |
The strictly positive shape parameter of the Weibull distribution ( |
delta |
The strictly positive parameter ( |
Details
The following is the probability density function of the Weibull distribution:
f(x)=\frac{\beta}{\alpha}\left(\frac{x}{\alpha}\right)^{\beta-1}e^{-(\frac{x}{\alpha})^{\beta}},
where x > 0, \alpha > 0 and \beta > 0.
Value
The functions se_wei, re_wei, hce_wei, and ae_wei provide the Shannon entropy, Rényi entropy, Havrda and Charvat entropy, and Arimoto entropy, respectively, depending on the selected parametric values of the Weibull distribution and \delta.
Author(s)
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.
References
Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of applied mechanics, 18, 293-297.
See Also
Examples
se_wei(1.2, 0.2)
delta <- c(1.5, 2, 3)
re_wei(1.2, 0.2, delta)
hce_wei(1.2, 0.2, delta)
ae_wei(1.2, 0.2, delta)