| Title: | Discrete Statistical Distributions |
| Version: | 1.1.3 |
| Description: | Implementation of new discrete statistical distributions. Each distribution includes the traditional functions as well as an additional function called the family function, which can be used to estimate parameters within the 'gamlss' framework. |
| License: | MIT + file LICENSE |
| Imports: | gamlss, pracma, Rcpp, COMPoissonReg, nleqslv |
| LinkingTo: | Rcpp |
| Encoding: | UTF-8 |
| Suggests: | knitr, rmarkdown |
| URL: | https://github.com/fhernanb/DiscreteDists |
| BugReports: | https://github.com/fhernanb/DiscreteDists/issues |
| Depends: | R (≥ 4.1) |
| LazyData: | true |
| Config/roxygen2/version: | 8.1.0 |
| NeedsCompilation: | yes |
| Packaged: | 2026-09-06 23:11:13 UTC; freddy |
| Author: | Freddy Hernandez-Barajas
 [aut, cre],
Fernando Marmolejo-Ramos
[aut],
Olga Usuga-Manco
[aut],
Jamiu Olumoh
[aut],
Osho Ajayi [aut] |
| Maintainer: | Freddy Hernandez-Barajas <fhernanb@unal.edu.co> |
| Repository: | CRAN |
| Date/Publication: | 2026-09-06 23:30:02 UTC |
Auxiliar function for hyper Poisson
Description
This function is used to calculate (a)r.
Usage
AR(a, r)
Arguments
a |
first value. |
r |
second value. |
Value
returns the value for the a(r) function.
The Bernoulli-geometric distribution
Description
The function BerG() defines the
Bernoulli-geometric distribution,
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
BerG(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The BerG distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \frac{(1-\mu+\sigma)}{(1+\mu+\sigma)} if x=0,
f(x | \mu, \sigma) = 4 \mu \frac{(\mu+\sigma-1)^{x-1}}{(\mu+\sigma+1)^{x+1}} if x=1, 2, ...,
with \mu > 0, \sigma > 0 and \sigma>|\mu-1|.
Value
Returns a gamlss.family object which can be used
to fit a BerG distribution
in the gamlss() function.
Author(s)
Hermes Marques, hermes.marques@ufrn.br
References
Bourguignon, M., & de Medeiros, R. M. (2022). A simple and useful regression model for fitting count data. Test, 31(3), 790-827.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rBerG(n=500, mu=0.75, sigma=0.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=BerG,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu and sigma
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ BerG
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
x4 <- runif(n)
mu <- exp(1 + 1.2*x1 + 0.2*x2)
sigma <- exp(2 + 1.5*x3 + 1.5*x4)
y <- rBerG(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2, x3=x3, x4=x4)
}
set.seed(16494786)
datos <- gendat(n=500)
mod2 <- gamlss(y~x1+x2, sigma.fo=~x3+x4, family=BerG, data=datos,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)
# Example using the dataset grazing from the bergreg package
# https://github.com/rdmatheus/bergreg
# This example corresponds to example 5.1
# presented by Bourguignon & Medeiros (2022)
# A simple and useful regression model for fitting count data
data("grazing")
hist(grazing$birds)
mod3 <- gamlss(birds ~ when + grazed,
sigma.fo=~1,
family=BerG, data=grazing,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod3)
The COMPO family
Description
The function COMPO() defines the
Conway-Maxwell-Poisson distribution,
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
COMPO(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The COMPO distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \frac{\mu^x}{(x!)^{\sigma} Z(\mu, \sigma)}
with \mu > 0, \sigma \geq 0 and
Z(\mu, \sigma)=\sum_{j=0}^{\infty} \frac{\mu^j}{(j!)^\sigma}.
The proposed functions here are based on the functions from the COMPoissonReg package.
Value
Returns a gamlss.family object which can be used
to fit a COMPO distribution
in the gamlss() function.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., & Boatwright, P. (2005). A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution. Journal of the Royal Statistical Society Series C: Applied Statistics, 54(1), 127-142.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
## Not run:
set.seed(12)
y <- rCOMPO(n=100, mu=10, sigma=3)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=COMPO,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
## End(Not run)
# Example 2
# Generating random values under some model
## Not run:
# A function to simulate a data set with Y ~ COMPO
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(2 + 1 * x1) # 12 approximately
sigma <- exp(2 - 2 * x2) # 2.71 approximately
y <- rCOMPO(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(123)
dat <- gendat(n=100)
# Fitting the model
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=COMPO, data=dat,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)
## End(Not run)
# Example 3
# Using the data from Shmueli et al. (2005) page 134
# The dataset consists of quarterly sales of a well-known brand of a
# particular article of clothing at stores of a large national retailer.
## Not run:
values <- 0:30
freq <- c(514, 503, 457, 423, 326, 233, 195, 139, 101, 77, 56, 40,
37, 22, 9, 7, 10, 9, 3, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1)
y <- rep(x=values, times=freq)
mod3 <- gamlss(y~1, sigma.fo=~1, family=COMPO,
control=gamlss.control(n.cyc=500, trace=TRUE))
exp(coef(mod3, what="mu"))
exp(coef(mod3, what="sigma"))
estim_mu_sigma_COMPO(y)
library(COMPoissonReg)
fit <- glm.cmp(y ~ 1)
res <- exp(fit$opt.res$par)
res
## End(Not run)
# Example 4
# Using the data from Shmueli et al. (2005) page 134
# The dataset contains lengths of words (numbers of syllables)
# in a Hungarian dictionary
## Not run:
# Slovak dictionary
y <- rep(x=1:5, times=c(7, 33, 49, 22, 6))
# Hungarian dictionary
y <- rep(x=1:9, times=c(1421, 12333, 20711, 15590, 5544, 1510, 289, 60, 1))
mod4 <- gamlss(y~1, sigma.fo=~1, family=COMPO,
control=gamlss.control(n.cyc=500, trace=TRUE))
exp(coef(mod4, what="mu"))
exp(coef(mod4, what="sigma"))
estim_mu_sigma_COMPO(y)
library(COMPoissonReg)
fit <- glm.cmp(y ~ 1)
res <- exp(fit$opt.res$par)
res
## End(Not run)
The COMPO2 family (with mu as mean)
Description
The function COMPO2() defines the
Conway-Maxwell-Poisson distribution
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
This parameterization was
proposed by Ribeiro et al. (2020) and the main
characteristic is that E(X)=\mu.
Usage
COMPO2(mu.link = "log", sigma.link = "identity")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "identity" link as the default for the sigma. |
Details
The COMPO2 distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \left(\mu + \frac{\exp(\sigma)-1}{2 \exp(\sigma)} \right)^{x \exp(\sigma)} \frac{(x!)^{\exp(\sigma)}}{Z(\mu, \sigma)}
with \mu > 0, \sigma \in \Re and
Z(\mu, \sigma)=\sum_{j=0}^{\infty} \frac{\mu^j}{(j!)^\sigma}.
The proposed functions here are based on the functions from the COMPoissonReg package.
Value
Returns a gamlss.family object which can be used
to fit a COMPO2 distribution
in the gamlss() function.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Ribeiro Jr, Eduardo E., et al. "Reparametrization of COM–Poisson regression models with applications in the analysis of experimental data." Statistical Modelling 20.5 (2020): 443-466.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rCOMPO2(n=500, mu=10, sigma=-1)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=COMPO2,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
coef(mod1, what="sigma")
# Example 2
# Generating random values under some model
## Not run:
# A function to simulate a data set with Y ~ COMPO2
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(2 + 1 * x1) # 12 approximately
sigma <- 1 - 2 * x2 # 0 approximately
y <- rCOMPO2(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(123)
dat <- gendat(n=200)
# Fitting the model
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=COMPO2, data=dat)
summary(mod2)
## End(Not run)
The Discrete Burr Hatke family
Description
The function DBH() defines the
Discrete Burr Hatke distribution
a single parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
DBH(mu.link = "logit")
Arguments
mu.link |
defines the mu.link, with "logit" link as the default for the mu parameter. Other links are "probit" and "cloglog"'(complementary log-log) |
Details
The Discrete Burr-Hatke distribution with parameter \mu has a support
0, 1, 2, ... and its probability mass function (pmf) is given by
f(x | \mu) = (\frac{1}{x+1}-\frac{\mu}{x+2})\mu^{x}
The pmf is log-convex for all values of 0 < \mu < 1, where \frac{f(x+1;\mu)}{f(x;\mu)}
is an increasing function in x for all values of the parameter \mu.
Note: in this implementation we changed the original parameters \lambda for \mu,
we did it to implement this distribution within gamlss framework.
Value
Returns a gamlss.family object which can be used
to fit a Discrete Burr-Hatke distribution
in the gamlss() function.
Author(s)
Valentina Hurtado Sepulveda, vhurtados@unal.edu.co
References
El-Morshedy, M., Eliwa, M. S., & Altun, E. (2020). Discrete Burr-Hatke distribution with properties, estimation methods and regression model. IEEE access, 8, 74359-74370.
See Also
dDBH.
Examples
# Example 1
# Generating some random values with
# known mu
y <- rDBH(n=1000, mu=0.74)
library(gamlss)
mod1 <- gamlss(y~1, family=DBH,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse logit function
inv_logit <- function(x) exp(x) / (1+exp(x))
inv_logit(coef(mod1, parameter="mu"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ DBH
gendat <- function(n) {
x1 <- runif(n)
mu <- inv_logit(-3 + 5 * x1)
y <- rDBH(n=n, mu=mu)
data.frame(y=y, x1=x1)
}
datos <- gendat(n=150)
mod2 <- NULL
mod2 <- gamlss(y~x1, family=DBH, data=datos,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
# Example 3
# Number of carious teeth among the four deciduous molars.
# Taken from EL-MORSHEDY (2020) page 74364.
y <- rep(0:4, times=c(64, 17, 10, 6, 3))
mod3 <- gamlss(y~1, family=DBH,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod3, what="mu"))
# Example 4
# Counts of cysts of kidneys using steroids.
# Taken from EL-MORSHEDY (2020) page 74365.
y <- rep(0:11, times=c(65, 14, 10, 6, 4, 2, 2, 2, 1, 1, 1, 2))
mod4 <- gamlss(y~1, family=DBH,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod4, what="mu"))
The DGEII distribution
Description
The function DGEII() defines the
Discrete generalized exponential distribution of a Second type
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
DGEII(mu.link = "logit", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "logit" link as the default for the mu parameter. Other links are "probit" and "cloglog"'(complementary log-log). |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The DGEII distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = (1-\mu^{x+1})^{\sigma}-(1-\mu^x)^{\sigma}
with 0 < \mu < 1 and \sigma > 0. If \sigma=1, the DGEII distribution
reduces to the geometric distribution with success probability 1-\mu.
Note: in this implementation we changed the original parameters
p to \mu and \alpha to \sigma,
we did it to implement this distribution within gamlss framework.
Value
Returns a gamlss.family object which can be used
to fit a DGEII distribution
in the gamlss() function.
Author(s)
Valentina Hurtado Sepúlveda, vhurtados@unal.edu.co
References
Nekoukhou, V., Alamatsaz, M. H., & Bidram, H. (2013). Discrete generalized exponential distribution of a second type. Statistics, 47(4), 876-887.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rDGEII(n=100, mu=0.75, sigma=0.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=DGEII,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ DGEII
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- inv_logit(1.7 - 2.8*x1)
sigma <- exp(0.73 + 1*x2)
y <- rDGEII(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
datos <- gendat(n=100)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=DGEII, data=datos,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)
# Example 3
# Number of accidents to 647 women working on H. E. Shells
# for 5 weeks. Taken from
# Nekoukhou V, Alamatsaz MH, Bidram H (2013) page 886.
y <- rep(x=0:5, times=c(447, 132, 42, 21, 3, 2))
mod3 <- gamlss(y~1, family=DGEII,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod3, what="mu"))
exp(coef(mod3, what="sigma"))
The discrete Inverted Kumaraswamy family
Description
The function DIKUM() defines the
discrete Inverted Kumaraswamy distribution
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
DIKUM(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The discrete Inverted Kumaraswamy distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and density given by
f(x | \mu, \sigma) = (1-(2+x)^{-\mu})^{\sigma}-(1-(1+x)^{-\mu})^{\sigma}
with \mu > 0 and \sigma > 0.
Note: in this implementation we changed the original parameters \alpha and \beta
for \mu and \sigma respectively, we did it to implement this distribution within gamlss framework.
Value
Returns a gamlss.family object which can be used
to fit a discrete Inverted Kumaraswamy distribution
in the gamlss() function.
Author(s)
Daniel Felipe Villa Rengifo, dvilla@unal.edu.co
References
El-Helbawy, A. A., Hegazy, M. A., Al-Dayian, G. R., & Abd EL-Kader, R. E. (2022). A discrete analog of the inverted Kumaraswamy distribution: properties and estimation with application to COVID-19 data. Pakistan Journal of Statistics and Operation Research, 18(1), 297-328.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(150)
y <- rDIKUM(1000, mu=1, sigma=5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y ~ 1, sigma.fo = ~1, family=DIKUM,
control = gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
library(gamlss)
# A function to simulate a data set with Y ~ DIKUM
gendat <- function(n) {
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(1.21 - 3 * x1) # 0.75 approximately
sigma <- exp(1.26 - 2 * x2) # 1.30 approximately
y <- rDIKUM(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
dat <- gendat(n=1000)
# Fitting the model
mod2 <- gamlss(y ~ x1, sigma.fo = ~x2, family = "DIKUM", data=dat,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
The Discrete Lindley family
Description
The function DLD() defines the
Discrete Lindley distribution
a single parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
DLD(mu.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
Details
The Discrete Lindley distribution with parameters \mu > 0 has a support
0, 1, 2, ... and density given by
f(x | \mu) = \frac{e^{-\mu x}}{1 + \mu} (\mu(1 - 2e^{-\mu}) + (1- e^{-\mu})(1+\mu x))
The parameter \mu can be interpreted as a strict upper bound on the failure rate function
The conventional discrete distributions (such as geometric, Poisson, etc.) are not suitable for various scenarios like reliability, failure times, and counts. Consequently, alternative discrete distributions have been created by adapting well-known continuous models for reliability and failure times. Among these, the discrete Weibull distribution stands out as the most widely used. But models like these require two parameters and not many of the known discrete distributions can provide accurate models for both times and counts, which the Discrete Lindley distribution does.
Note: in this implementation we changed the original parameters \theta for \mu,
we did it to implement this distribution within gamlss framework.
Value
Returns a gamlss.family object which can be used
to fit a Discrete Lindley distribution
in the gamlss() function.
Author(s)
Yojan Andrés Alcaraz Pérez, yalcaraz@unal.edu.co
References
Bakouch, H. S., Jazi, M. A., & Nadarajah, S. (2014). A new discrete distribution. Statistics, 48(1), 200-240.
See Also
dDLD.
Examples
# Example 1
# Generating some random values with
# known mu
y <- rDLD(n=100, mu=0.3)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=DLD,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod1, what="mu"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ DLD
gendat <- function(n) {
x1 <- runif(n)
mu <- exp(2 - 4 * x1)
y <- rDLD(n=n, mu=mu)
data.frame(y=y, x1=x1)
}
set.seed(1235)
datos <- gendat(n=150)
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=DLD, data=datos,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
# Example 3
# Survival times in days of 72 guinea pigs.
# Taken from Bakouch et al (2014) page 26.
y <- c(12, 15, 22, 24, 24, 32, 32, 33, 34, 38, 38, 43, 44, 48,
52, 53, 54, 54, 55, 56, 57, 58, 58, 59, 60, 60, 60, 60,
61, 62, 63, 65, 65, 67, 68, 70, 70, 72, 73, 75, 76, 76,
81, 83, 84, 85, 87, 91, 95, 96, 98, 99, 109, 110, 121,
127, 129, 131, 143, 146, 146, 175, 175, 211, 233, 258,
258, 263, 297, 341, 341, 376)
mod3 <- gamlss(y~1, family=DLD,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod3, what="mu"))
# Example 4
# Remission times in weeks for 20 leukaemia patients.
# Taken from Bakouch et al (2014) page 26.
y <- c(1, 3, 3, 6, 7, 7, 10, 12, 14, 15, 18, 19,
22, 26, 28, 29, 34, 40, 48, 49)
mod4 <- gamlss(y~1, family=DLD,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod4, what="mu"))
# Example 5
# Numbers of fires in Greece for the period from 1
# July 1998 to 31 August of the same year .
# Taken from Bakouch et al (2014) page 26.
y <- c(2, 4, 4, 3, 3, 1, 2, 4, 3, 1, 1, 0, 5, 5, 0, 3, 1,
1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 4, 2, 2, 1, 2, 1, 2, 0, 2, 2, 1, 0, 3, 2, 1, 2,
2, 7, 3, 5, 2, 5, 4, 5, 6, 5, 4, 3, 8, 43, 8, 4, 4,
3, 10, 5, 4, 5, 12, 3, 8, 12, 10, 11, 6, 1, 8, 9,
12, 9, 4, 8, 12, 11, 8, 6, 4, 7, 9, 15, 12, 15, 15,
12, 9, 16, 7, 11, 9, 11, 6, 5, 20, 9, 8, 8, 5, 7, 10,
6, 6, 5, 5, 15, 6, 8, 5, 6)
mod5 <- gamlss(y~1, family=DLD,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod5, what="mu"))
# Example 6
# Final examination marks of space students in
# mathematics in the Indian Institute of Technology at Kanpur.
# Taken from Bakouch et al (2014) page 26.
y <- c(29, 25, 50, 15, 13, 27, 15, 18, 7, 7, 8, 19, 12,
18, 5, 21, 15, 86, 21, 15, 14, 39, 15, 14, 70,
44, 6, 23, 58, 19, 50, 23, 11, 6, 34, 18, 28, 34,
12, 37, 4, 60, 20, 23, 40, 65, 19, 31)
mod6 <- gamlss(y~1, family=DLD,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod6, what="mu"))
The DMOLBE family
Description
The function DMOLBE() defines the
Discrete Marshall-Olkin Length Biased Exponential distribution
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
DMOLBE(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The DMOLBE distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \frac{\sigma ((1+x/\mu)\exp(-x/\mu)-(1+(x+1)/\mu)\exp(-(x+1)/\mu))}{(1-(1-\sigma)(1+x/\mu)\exp(-x/\mu)) ((1-(1-\sigma)(1+(x+1)/\mu)\exp(-(x+1)/\mu))}
with \mu > 0 and \sigma > 0
Value
Returns a gamlss.family object which can be used
to fit a DMOLBE distribution
in the gamlss() function.
Author(s)
Olga Usuga, olga.usuga@udea.edu.co
References
Aljohani, H. M., Ahsan-ul-Haq, M., Zafar, J., Almetwally, E. M., Alghamdi, A. S., Hussam, E., & Muse, A. H. (2023). Analysis of Covid-19 data using discrete Marshall–Olkinin length biased exponential: Bayesian and frequentist approach. Scientific Reports, 13(1), 12243.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rDMOLBE(n=200, mu=3, sigma=7)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=DMOLBE)
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ DMOLBE
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1.21 - 3 * x1) # 0.75 approximately
sigma <- exp(1.26 - 2 * x2) # 1.30 approximately
y <- rDMOLBE(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(123)
dat <- gendat(n=200)
# Fitting the model
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=DMOLBE, data=dat,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
# Example 3
# Data Set I (death due to coronavirus in China). The first data set is the number
# of deaths due to coronavirus in China from 23 January to 28 March.
# The data sets used in the paper was collected from 2020 year. The data set
# is reported in https://www.worldometers.info/coronavirus/country/china/.
# The data are:
y <- c(8, 16, 15, 24, 26, 26, 38, 43, 46, 45, 57, 64, 65, 73, 73, 86, 89, 97,
108, 97, 146, 121, 143, 142, 105, 98, 136, 114, 118, 109, 97, 150, 71,
52, 29, 44, 47, 35, 42, 31, 38, 31, 30, 28, 27, 22, 17, 22, 11, 7,
13, 10, 14, 13, 11, 8, 3, 7, 6, 9, 7, 4, 6, 5, 3, 5)
# Fitting the model
mod3 <- gamlss(y~1, sigma.fo=~1, family=DMOLBE,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod3)
# Extracting the fitted values for mu and sigma
# using the inverse link function
mu_hat <- exp(coef(mod3, what="mu"))
mu_hat
sigma_hat <- exp(coef(mod3, what="sigma"))
sigma_hat
# Example 4
# Data Set II (daily death due to coronavirus in Pakistan). The second data
# set is the daily deaths due to coronavirus in Pakistan from 18 March
# to 30 June. The data sets used in the paper was collected from 2020 year.
# The data is reported in
# https://www.worldometers.info/coronavirus/country/Pakistan.
# The data are:
y <- c(1, 6, 6, 4, 4, 4, 1, 20, 5, 2, 3, 15, 17, 7, 8, 25, 8, 25, 11,
25, 16, 16, 12, 11, 20, 31, 42, 32, 23, 17, 19, 38, 50, 21, 14,
37, 23, 47, 31, 24, 9, 64, 39, 30, 36, 46, 32, 50, 34, 32, 34,
30, 28, 35, 57, 78, 88, 60, 78, 67, 82, 68, 97, 67, 65, 105,
83, 101, 107, 88, 178, 110, 136, 118, 136, 153, 119, 89, 105,
60, 148, 59, 73, 83, 49, 137, 91)
# Fitting the model
mod4 <- gamlss(y~1, sigma.fo=~1, family=DMOLBE,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod4)
# Extracting the fitted values for mu and sigma
# using the inverse link function
mu_hat <- exp(coef(mod4, what="mu"))
mu_hat
sigma_hat <- exp(coef(mod4, what="sigma"))
sigma_hat
The Discrete Perks family
Description
The function DPERKS() defines the
Discrete Perks distribution
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
DPERKS(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The discrete Perks distribution with parameters \mu > 0 and \sigma > 0
has a support 0, 1, 2, ... and density given by
f(x | \mu, \sigma) = \frac{\mu(1+\mu)(e^\sigma-1)e^{\sigma x}}{(1+\mu e^{\sigma x})(1+\mu e^{\sigma(x+1)})}
Note: in this implementation we changed the original parameters
\lambda for \mu and \beta for \sigma,
we did it to implement this distribution within gamlss framework.
Author(s)
Veronica Seguro Varela, vseguro@unal.edu.co
References
Tyagi, A., Choudhary, N., & Singh, B. (2020). A new discrete distribution: Theory and applications to discrete failure lifetime and count data. J. Appl. Probab. Statist, 15, 117-143.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123)
y <- rDPERKS(n=1000, mu=2.5, sigma=0.4)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=DPERKS,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ DPERKS
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(-1.6 + 5 * x1)
sigma <- exp(1.7 - 5 * x2)
y <- rDPERKS(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(12345)
datos <- gendat(n=1000)
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=DPERKS, data=datos,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)
# Example 3
# The dataset comes from Tyagi et al. (2020) page 21
# The dataset contains the number of outbreaks of strikes in the
# UK coal mining industries in four successive week periods
# in the year 1948-59.
y <- rep(0:4, times=c(46, 76, 24, 9, 1))
# Fitting the model
library(gamlss)
mod3 <- gamlss(y~1, family=DPERKS,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod3, what="mu"))
exp(coef(mod3, what="sigma"))
# Example 4
# The dataset comes from Tyagi et al. (2020) page 21
# The dataset contains the number fishes caught
# in traps (David, 1971, pg. 168).
y <- rep(0:9, times=c(1, 2, 11, 20, 29, 23, 10, 3, 1, 0))
# Fitting the model
library(gamlss)
mod4 <- gamlss(y~1, family=DPERKS,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod4, what="mu"))
exp(coef(mod4, what="sigma"))
# Example 5
# The dataset comes from Tyagi et al. (2020) page 24
# This dataset consists of remission times in weeks
# for 20 leukemia patients randomly assigned to a certain treatment
y <- c(1, 3, 3, 6, 7, 7, 10, 12, 14, 15, 18,
19, 22, 26, 28, 29, 34, 40, 48, 49)
# Fitting the model
library(gamlss)
mod4 <- gamlss(y~1, family=DPERKS)
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod4, what="mu"))
exp(coef(mod4, what="sigma"))
Discrete power-Ailamujia distribution
Description
The function DsPA() defines the
discrete power-Ailamujia distribution
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
DsPA(mu.link = "log", sigma.link = "logit")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "logit" link as the default for the sigma parameter. Other links are "probit" and "cloglog"'(complementary log-log). |
Details
The discrete power-Ailamujia distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and density given by
f(x | \mu, \sigma) = (\sigma^x)^\mu (1-x^\mu \log(\lambda)) - (\sigma^{(x+1)})^\mu (1-(x+1)^\mu \log(\lambda))
Note: in this implementation we changed the original parameters \beta and \lambda
for \mu and \sigma respectively, we did it to implement this distribution within gamlss framework.
Value
Returns a gamlss.family object which can be used
to fit a discrete power-Ailamujia distribution
in the gamlss() function.
Author(s)
Maria Camila Mena Romana, mamenar@unal.edu.co
References
Alghamdi, A. S., Ahsan-ul-Haq, M., Babar, A., Aljohani, H. M., Afify, A. Z., & Cell, Q. E. (2022). The discrete power-Ailamujia distribution: properties, inference, and applications. AIMS Math, 7(5), 8344-8360.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rDsPA(n=100, mu=1.2, sigma=0.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=DsPA,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
exp(coef(mod1, what="mu"))
inv_logit(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ DsPA
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
x4 <- runif(n)
mu <- exp(1 + 1.2*x1 + 0.2*x2)
sigma <- inv_logit(2 + 1.5*x3 + 1.5*x4)
y <- rDsPA(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2, x3=x3, x4=x4)
}
set.seed(123)
datos <- gendat(n=100)
mod2 <- gamlss(y~x1+x2, sigma.fo=~x3+x4, family=DsPA, data=datos,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)
# Example 3
# failure times for a sample of 15 electronic components in an acceleration life test
# Taken from
# Alghamdi et. al (202) page 8354.
y <- c(1.0, 5.0, 6.0, 11.0, 12.0, 19.0, 20.0, 22.0,
23.0, 31.0, 37.0, 46.0, 54.0, 60.0, 66.0)
mod3 <- gamlss(y~1, family=DsPA,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
exp(coef(mod3, what="mu"))
inv_logit(coef(mod3, what="sigma"))
# Example 4
# number of fires in Greece from July 1, 1998 to August 31, 1998.
# Taken from
# Alghamdi et. al (202) page 8354.
y <- c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4,
4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5,
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6,
6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7,
8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9,
9, 9, 9, 9, 10, 10, 10, 11, 11,
11, 11, 12, 12, 12, 12, 12, 12,
15, 15, 15, 15, 16, 20, 43)
mod4 <- gamlss(y~1, family=DsPA,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
exp(coef(mod4, what="mu"))
inv_logit(coef(mod4, what="sigma"))
Auxiliar function for hyper Poisson
Description
This function is used inside density function of Hyper Poisson.
Usage
F11(z, c, maxiter_series = 10000, tol = 1e-10)
Arguments
z, c |
values for F11. |
maxiter_series |
maximum value to obtain F11. |
tol |
this is the tolerance of the infinite sum. |
Value
returns the value for the F11 function.
The GGEO family
Description
The function GGEO() defines the
Generalized Geometric distribution
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
GGEO(mu.link = "logit", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "logit" link as the default for the sigma. Other links are "probit" and "cloglog"'(complementary log-log) |
Details
The GGEO distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \frac{\sigma \mu^x (1-\mu)}{(1-(1-\sigma) \mu^{x+1})(1-(1-\sigma) \mu^{x})}
with 0 < \mu < 1 and \sigma > 0. If \sigma=1, the GGEO distribution
reduces to the geometric distribution with success probability 1-\mu.
Value
Returns a gamlss.family object which can be used
to fit a GGEO distribution
in the gamlss() function.
Author(s)
Valentina Hurtado Sepúlveda, vhurtados@unal.edu.co
References
Gómez-Déniz, E. (2010). Another generalization of the geometric distribution. Test, 19, 399-415.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123)
y <- rGGEO(n=200, mu=0.95, sigma=1.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=GGEO,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ GGEO
## Not run:
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- inv_logit(1.7 - 2.8*x1)
sigma <- exp(0.73 + 1*x2)
y <- rGGEO(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(78353)
datos <- gendat(n=100)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=GGEO, data=datos,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)
## End(Not run)
# Example 3
# Number of accidents to 647 women working on H. E. Shells
# for 5 weeks. Taken from Gomez-Deniz (2010) page 411.
y <- rep(x=0:5, times=c(447, 132, 42, 21, 3, 2))
mod3 <- gamlss(y~1, family=GGEO,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod3, what="mu"))
exp(coef(mod3, what="sigma"))
# Example 4
# Total number of carious teeth among the four deciduous molars
# Taken from Gomez-Deniz (2010) page 412.
y <- rep(x=0:4, times=c(64, 17, 10, 6, 3))
mod4 <- gamlss(y~1, family=GGEO,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod4, what="mu"))
exp(coef(mod4, what="sigma"))
# Example 5
# Results of ten shots fired from a rifle at each of 100 targets.
# Taken from Gomez-Deniz (2010) page 412.
y <- rep(x=0:10, times=c(0, 2, 4, 10, 22, 26, 18, 12, 4, 2, 0))
mod5 <- gamlss(y~1, family=GGEO,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod5, what="mu"))
exp(coef(mod5, what="sigma"))
# Example 6
# Fish catch data in Kemp (1992).
# Taken from Gomez-Deniz (2010) page 412.
y <- rep(x=0:9, times=c(1, 2, 11, 20, 29, 23, 10, 3, 1, 0))
mod6 <- gamlss(y~1, family=GGEO,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod6, what="mu"))
exp(coef(mod6, what="sigma"))
The hyper Poisson family
Description
The function HYPERPO() defines the
hyper Poisson distribution
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
HYPERPO(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The hyper-Poisson distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and density given by
f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}
where the function _1F_1(a;c;z) is defined as
_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}
and (a)_r = \frac{\gamma(a+r)}{\gamma(a)} for a>0 and r positive integer.
Note: in this implementation we changed the original parameters \lambda and \gamma
for \mu and \sigma respectively, we did it to implement this distribution within gamlss framework.
Value
Returns a gamlss.family object which can be used
to fit a hyper-Poisson distribution
in the gamlss() function.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Sáez-Castillo, A. J., & Conde-Sánchez, A. (2013). A hyper-Poisson regression model for overdispersed and underdispersed count data. Computational Statistics & Data Analysis, 61, 148-157.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rHYPERPO(n=200, mu=10, sigma=1.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=HYPERPO,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ HYPERPO
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1.21 - 3 * x1) # 0.75 approximately
sigma <- exp(1.26 - 2 * x2) # 1.30 approximately
y <- rHYPERPO(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(1234)
dat <- gendat(n=100)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=HYPERPO, data=dat,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)
The hyper Poisson family (with mu as mean)
Description
The function HYPERPO2() defines the
hyper Poisson distribution (with mu as mean)
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
HYPERPO2(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The hyper-Poisson distribution with parameters \mu and \sigma
has a support 0, 1, 2, ...
Note: in this implementation the parameter \mu is the mean
of the distribution and \sigma corresponds to
the dispersion parameter. If you fit a model with this parameterization,
the time will increase because an internal procedure to convert \mu
to \lambda parameter.
Value
Returns a gamlss.family object which can be used
to fit a hyper-Poisson distribution version 2
in the gamlss() function.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Sáez-Castillo, A. J., & Conde-Sánchez, A. (2013). A hyper-Poisson regression model for overdispersed and underdispersed count data. Computational Statistics & Data Analysis, 61, 148-157.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rHYPERPO2(n=100, mu=4, sigma=1.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=HYPERPO2,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
## Not run:
# A function to simulate a data set with Y ~ HYPERPO2
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1.21 - 3 * x1) # 0.75 approximately
sigma <- exp(1.26 - 2 * x2) # 1.30 approximately
y <- rHYPERPO2(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(12345)
dat <- gendat(n=200)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=HYPERPO2, data=dat,
control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)
## End(Not run)
NMES1988 dataset
Description
Cross-section data originating from the US National Medical Expenditure Survey (NMES) conducted in 1987 and 1988. The NMES is based upon a representative, national probability sample of the civilian non-institutionalized population and individuals admitted to long-term care facilities during 1987. The data are a subsample of individuals ages 66 and over all of whom are covered by Medicare (a public insurance program providing substantial protection against health-care costs).
Usage
NMES1988
Format
NMES1988
A data frame containing 4,406 observations on 19 variables:
- visits
Number of physician office visits.
- nvisits
Number of non-physician office visits.
- ovisits
Number of physician hospital outpatient visits.
- novisits
Number of non-physician hospital outpatient visits.
- emergency
Emergency room visits.
- hospital
Number of hospital stays.
- health
Factor indicating self-perceived health status, levels are "poor", "average" (reference category), "excellent".
- chronic
Number of chronic conditions.
- adl
Factor indicating whether the individual has a condition that limits activities of daily living ("limited") or not ("normal").
- region
Factor indicating region, levels are northeast, midwest, west, other (reference category).
- age
Age in years (divided by 10).
- afam
Factor. Is the individual African-American?
- gender
Factor indicating gender.
- married
Factor. Is the individual married?
- school
Number of years of education.
- income
Family income in USD 10,000.
- employed
Factor. Is the individual employed?
- insurance
Factor. Is the individual covered by private insurance?
- medicaid
Factor. Is the individual covered by Medicaid?
Author(s)
AER R package.
Source
This dataset comes from the AER R package.
New Poisson-generalised Lindley distribution
Description
The function NPGL() defines the
Poisson-generalised Lindley distribution,
a two parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
NPGL(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
Details
The Poisson-generalised Lindley distribution with parameters \mu and \sigma
has support x = 0, 1, 2, \ldots and probability mass function given by
f(x \mid \mu, \sigma)=\frac{\mu^2+\frac{\mu^{\sigma}(\mu+1)^{1-\sigma}\Gamma(x+\sigma)}{\Gamma(\sigma)\Gamma(x+1)}}{(\mu+1)^{x+2}}
with \mu > 0 and \sigma > 0.
This distribution is useful for modeling over-dispersed count data.
Note: in this implementation we changed the original parameters \theta and \alpha
for \mu and \sigma respectively, we did it to implement this distribution within gamlss framework.
Value
Returns a gamlss.family object which can be used
to fit a NPGL distribution
in the gamlss() function.
Author(s)
Tomás Mesa, tomas.mesaz@udea.edu.co
References
Altun, E. A new two-parameter discrete poisson-generalized Lindley distribution with properties and applications to healthcare data sets. Comput Stat 36, 2841–2861 (2021). https://doi.org/10.1007/s00180-021-01097-0
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123)
y <- rNPGL(n=100, mu=20, sigma=2)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=NPGL,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu and sigma
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ NPGL
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1.7 - 2.8 * x1) # Approx 1.35
sigma <- exp(0.73 + 1 * x2) # Approx 3.42
y <- rNPGL(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(1234)
datos <- gendat(n=200)
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=NPGL, data=datos,
control=gamlss.control(n.cyc=800, trace=FALSE))
summary(mod2)
Mean-parametrized New Poisson-generalised Lindley distribution (NPGL2)
Description
The function NPGL2() defines the mean-parametrized version of the
Poisson-generalised Lindley distribution, a two-parameter distribution,
for a gamlss.family object to be used in GAMLSS fitting using
the function gamlss().
Usage
NPGL2(mu.link = "log", sigma.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter (the mean of the distribution). |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma parameter (parameter alpha of the original NPGL). |
Details
This family uses the mean-parametrized version of the NPGL distribution proposed in Section 6 of Altun (2021). The new parameters are:
-
\mu > 0: the mean of the distribution, i.e.E[X]. -
\sigma > 0
The reparametrization links the mean and \sigma
to the original parameter \mu via the mean
equation E[X] = (\sigma + \mu) / (\mu (1 + \mu))
Value
Returns a gamlss.family object which can be used to fit a
mean-parametrized NPGL distribution in the gamlss() function.
Author(s)
Tomas Mesa, tomas.mesaz@udea.edu.co
References
Altun, E. A new two-parameter discrete poisson-generalized Lindley distribution with properties and applications to healthcare data sets. Comput Stat 36, 2841-2861 (2021). https://doi.org/10.1007/s00180-021-01097-0
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123)
y <- rNPGL2(n=5000, mu=4, sigma=2)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=NPGL2,
control=gamlss.control(n.cyc=500, trace=TRUE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ NPGL2
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1.5 - 1.8 * x1) # mean of the distribution
sigma <- exp(0.5 + 0.8 * x2) # shape parameter alpha
y <- rNPGL2(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(123)
dat <- gendat(n=1000)
# Fitting the model
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=NPGL2, data=dat,
control=gamlss.control(n.cyc=800, trace=TRUE))
summary(mod2)
# Example 3
# Using the data from Altun (2021), Section 8.1.
# The dataset is about the 1991 Arizona cardiovascular patient.
# We model the length of hospital stay (los) with cardiovascular
# procedure, sex, type of admission and age as covariates.
# The response variable has a dispersion index of 5.43,
# confirming over-dispersion.
# Data available in the azpro dataset of the COUNT package.
mod3 <- gamlss(
los ~ procedure + sex + admit + age75,
sigma.fo = ~1,
family = NPGL2(mu.link = "log"),
data = azpro,
control = gamlss.control(n.cyc = 500, trace = TRUE)
)
# Extracting the fitted values for mu and sigma
# using the inverse link function
coef(mod3, what="mu")
coef(mod3, what="sigma")
# Note: coef(mod3, what="mu") is the estimated mean length
# of hospital stay for the reference group (PTCA procedure,
# female, elective admission, age <= 75), directly interpretable
# because mu is the mean of the NPGL2 distribution.
# Example 4
# Using the data from Altun (2021), Section 8.2.
# The dataset originates from the US National Medical Expenditure
# Survey (NMES) conducted in 1987 and 1988. We model the number
# of physician office visits with hospital stays, chronic conditions,
# activity limitations, age, gender, marital status, income,
# employment status, Medicaid, private insurance and self-perceived
# health status as covariates.
# The response variable has a dispersion index of 7.91,
# confirming over-dispersion.
# Data available in the NMES1988 dataset of the AER package.
# Set "average" as the reference category for health status,
# so that healthexcellent and healthpoor are the two dummies,
# matching x11 and x12 in Altun (2021).
NMES1988$health <- relevel(factor(NMES1988$health), ref = "average")
mod4 <- gamlss(
visits ~ hospital + chronic + adl + age + gender + married +
income + employed + medicaid + insurance + health,
sigma.fo = ~1,
family = NPGL2(mu.link = "log"),
data = NMES1988,
control = gamlss.control(n.cyc = 500, trace = TRUE)
)
coef(mod4, what="mu")
coef(mod4, what="sigma")
The Discrete Poisson XLindley
Description
The function POISXL() defines the
Discrete Poisson XLindley distribution
a single parameter distribution,
for a gamlss.family object to be used in GAMLSS
fitting using the function gamlss().
Usage
POISXL(mu.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
Details
The Discrete Poisson XLindley distribution with parameters \mu has a support
0, 1, 2, ... and mass function given by
f(x | \mu) = \frac{\mu^2(x+\mu^2+3(1+\mu))}{(1+\mu)^{4+x}}; with \mu>0.
Note: in this implementation we changed the original parameters \alpha for \mu,
we did it to implement this distribution within gamlss framework.
Value
Returns a gamlss.family object which can be used
to fit a Discrete Poisson XLindley distribution
in the gamlss() function.
Author(s)
Mariana Blandon Mejia, mblandonm@unal.edu.co
References
Ahsan-ul-Haq, M., Al-Bossly, A., El-Morshedy, M., & Eliwa, M. S. (2022). Poisson XLindley distribution for count data: statistical and reliability properties with estimation techniques and inference. Computational Intelligence and neuroscience, 2022(1), 6503670.
See Also
Examples
# Example 1
# Generating some random values with
# known mu
y <- rPOISXL(n=1000, mu=1)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=POISXL,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod1, what="mu"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ POISXL
gendat <- function(n) {
x1 <- runif(n, min=0.4, max=0.6)
mu <- exp(1.21 - 3 * x1) # 0.75 approximately
y <- rPOISXL(n=n, mu=mu)
data.frame(y=y, x1=x1)
}
dat <- gendat(n=1500)
# Fitting the model
mod2 <- NULL
mod2 <- gamlss(y~x1, family=POISXL, data=dat,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
# Example 3
# The counts the number of borers per hill of corn in an
# experiment conducted randomly on 8 hills in 15 replications.
# Taken from Ahsan-ul-Haq et al (2022) page 10.
y <- rep(x=0:8, times=c(43, 35, 17, 11, 5, 4, 1, 2, 2))
mod3 <- gamlss(y~1, family=POISXL,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
exp(coef(mod3, what="mu"))
# Example 4
# The number of mammalian cytogenetic dosimetry lesions produced
# by streptogramin exposure in rabbit.
# Taken from Ahsan-ul-Haq et al (2022) page 10.
y <- rep(x=0:4, times=c(200, 57, 30, 7, 6))
mod4 <- gamlss(y~1, family=POISXL,
control=gamlss.control(n.cyc=500, trace=FALSE))
# Extracting the fitted values for mu
exp(coef(mod4, what="mu"))
Poisson-transmuted record type exponential distribution
Description
The function PTRTE() defines the Poisson-transmuted record type exponential distribution,
a two-parameter discrete distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
Usage
PTRTE(mu.link = "log", sigma.link = "logit")
Arguments
mu.link |
defines the link function for the mu parameter, with "log" as the default. |
sigma.link |
defines the link function for the sigma parameter, with "logit" as the default. |
Details
The Poisson-transmuted record type exponential distribution with parameters \mu
and \sigma has support x = 0,1,2,\dots and probability mass function given by
f(x | \mu, \sigma) = \frac{\mu}{(1+\mu)^{x+1}} \left(\frac{\sigma \mu (1+x)}{1+\mu} - (\sigma-1) \right)
Parameter restrictions:
\mu > 0 and 0 < \sigma < 1.
Note: we renamed the original parameters \theta and p to \mu and \sigma respectively
to implement this distribution within the gamlss framework.
Value
Returns a gamlss.family object that can be used to fit the Poisson-transmuted record type exponential
distribution using the gamlss() function.
Author(s)
Rebeca Isabel Rodriguez Gonzalez, rebeca.rodriguez@udea.edu.co
References
Erbayram, T., & Akdogan, Y. (2025). A new discrete model generated from mixed Poisson transmuted record type exponential distribution. Ricerca di Matematica, 74, 1225–1247.
See Also
Examples
# Example 1
# Generating some random values with known mu and sigma
# logit_inv function
logit_inv <- function(x) exp(x) / (exp(x)+1)
set.seed(12345)
y <- rPTRTE(n=200, mu=0.2, sigma=0.5)
# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=PTRTE)
# Extracting the fitted values for mu and sigma
exp(coef(mod1, what="mu"))
logit_inv(coef(mod1, what="sigma"))
# Example 2
# Generating random values under some model
# A function to simulate a data set with Y ~ PTRTE
gendat <- function(n) {
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(2 + 1 * x1) # 12 approximately
sigma <- logit_inv(2 - 2 * x2) # 0.73 approximately
y <- rPTRTE(n=n, mu=mu, sigma=sigma)
data.frame(y=y, x1=x1, x2=x2)
}
set.seed(12345)
dat <- gendat(n=2000)
# Fitting the model
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, family=PTRTE, data=dat,
control=gamlss.control(n.cyc=500, trace=FALSE))
summary(mod2)
# Example 3 (Second data set of the article)
# European corn-borer count data reported by McGuire et al. (1957).
# The observed and fitted frequencies are given in Table 11 of
# Erbayram and Akdogan (2025), where the P-TRTE distribution is
# illustrated using this data set.
values <- 0:5
freq <- c(188, 83, 36, 14, 2, 1)
y <- rep(x=values, times=freq)
mod3 <- gamlss(y~1, sigma.fo=~1, family=PTRTE(),
control=gamlss.control(n.cyc=500, trace=TRUE))
exp(coef(mod3, what="mu"))
logit_inv(coef(mod3, what="sigma"))
Sum of One-Dimensional Functions
Description
Sum of One-Dimensional Functions
Usage
add(f, lower, upper, ..., abs.tol = .Machine$double.eps)
Arguments
f |
an R function taking a numeric first argument and returning a numeric vector of the same length. |
lower |
the lower limit of sum. Can be infinite. |
upper |
the upper limit of sum. Can be infinite. |
... |
additional arguments to be passed to f. |
abs.tol |
absolute accuracy requested. |
Value
This function returns the sum value.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
Examples
# Poisson expected value
add(f=function(x, lambda) x*dpois(x, lambda), lower=0, upper=Inf,
lambda=7.5)
# Binomial expected value
add(f=function(x, size, prob) x*dbinom(x, size, prob), lower=0, upper=20,
size=20, prob=0.5)
# Examples with infinite series
add(f=function(x) 0.5^x, lower=0, upper=100) # Ans=2
add(f=function(x) (1/3)^(x-1), lower=1, upper=Inf) # Ans=1.5
add(f=function(x) 4/(x^2+3*x+2), lower=0, upper=Inf, abs.tol=0.001) # Ans=4.0
add(f=function(x) 1/(x*(log(x)^2)), lower=2, upper=Inf, abs.tol=0.000001) # Ans=2.02
add(f=function(x) 3*0.7^(x-1), lower=1, upper=Inf) # Ans=10
add(f=function(x, a, b) a*b^(x-1), lower=1, upper=Inf, a=3, b=0.7) # Ans=10
add(f=function(x, a=3, b=0.7) a*b^(x-1), lower=1, upper=Inf) # Ans=10
azpro dataset
Description
Data come from the 1991 Arizona cardiovascular patient files. A subset of the fields was selected to model the differential length of stay for patients entering the hospital to receive one of two standard cardiovascular procedures: CABG and PTCA. CABG is the standard acronym for Coronary Artery Bypass Graft, where the flow of blood in a diseased or blocked coronary artery or vein has been grafted to bypass the diseased sections. PTCA, or Percutaneous Transluminal Coronary Angioplasty, is a method of placing a balloon in a blocked coronary artery to open it to blood flow. It is a much less severe method of treatment for those having coronary blockage, with a corresponding reduction in risk.
Usage
azpro
Format
azpro
A data frame with 3589 observations on the following 6 variables:
- los
length of hospital stay
- procedure
1=CABG; 0=PTCA
- sex
1=Male; 0=female
- admit
1=Urgent/Emerg; 0=elective (type of admission)
- age75
1=Age>75; 0=Age<=75
- hospital
encrypted facility code (string)
Author(s)
COUNT R package.
Source
This dataset comes from the COUNT R package.
Set the Right Link Function for Specified Parameter and Distribution
Description
This function is used within the distribution family specification of a GAMLSS model to define the right link for each of the parameters of the distribution. This function should not be called by the user unless he/she specify a new distribution family or wishes to change existing link functions in the parameters.
Usage
checklink(which.link = NULL, which.dist = NULL, link = NULL, link.List = NULL)
Arguments
which.link |
which parameter link e.g. |
which.dist |
which distribution family e.g. |
link |
a repetition of |
link.List |
what link function are required e.g. |
Details
This function is the same checklink fuction from gamlss.dist package.
Value
Defines the right link for each parameter.
Bernoulli-geometric distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the Bernoulli-geometric distribution
with parameters \mu and \sigma.
Usage
dBerG(x, mu, sigma, log = FALSE)
pBerG(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rBerG(n, mu, sigma)
qBerG(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The BerG distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \frac{(1-\mu+\sigma)}{(1+\mu+\sigma)} if x=0,
f(x | \mu, \sigma) = 4 \mu \frac{(\mu+\sigma-1)^{x-1}}{(\mu+\sigma+1)^{x+1}} if x=1, 2, ...,
with \mu > 0, \sigma > 0 and \sigma>|\mu-1|.
Value
dBerG gives the density, pBerG gives the distribution
function, qBerG gives the quantile function, rBerG
generates random deviates.
Author(s)
Hermes Marques, hermes.marques@ufrn.br
References
Bourguignon, M., & de Medeiros, R. M. (2022). A simple and useful regression model for fitting count data. Test, 31(3), 790-827.
See Also
BerG.
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 20
probs1 <- dBerG(x=0:x_max, mu=0.7, sigma=0.5)
probs2 <- dBerG(x=0:x_max, mu=0.3, sigma=1)
probs3 <- dBerG(x=0:x_max, mu=2, sigma=3)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for BerG",
ylim=c(0, 0.80))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=0.7, sigma=0.5",
"mu=0.3, sigma=1",
"mu=2, sigma=3"))
# Example 2
# Checking if the cumulative curves converge to 1
#plot1
x_max <- 10
plot_discrete_cdf(x=0:x_max,
fx=dBerG(x=0:x_max, mu=1, sigma=2),
col="dodgerblue",
main="CDF for BerG",
lwd=3)
legend("bottomright", legend="mu=1, sigma=2",
col="dodgerblue", lty=1, lwd=2, cex=0.8)
#plot2
plot_discrete_cdf(x=0:x_max,
fx=dBerG(x=0:x_max, mu=3, sigma=3),
col="tomato",
main="CDF for BerG",
lwd=3)
legend("bottomright", legend="mu=3, sigma=3",
col="tomato", lty=1, lwd=2, cex=0.8)
#plot3
plot_discrete_cdf(x=0:x_max,
fx=dBerG(x=0:x_max, mu=5, sigma=5),
col="green4",
main="CDF for BerG",
lwd=3)
legend("bottomright", legend="mu=5, sigma=5",
col="green4", lty=1, lwd=2, cex=0.8)
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 15
probs1 <- dBerG(x=0:x_max, mu=0.5, sigma=5)
names(probs1) <- 0:x_max
x <- rBerG(n=1000, mu=0.5, sigma=5)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 1
sigma <- 2
p <- seq(from=0, to=1, by=0.01)
qxx <- qBerG(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of DBerG(mu=1, sigma=2)")
The COMPO distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the
Conway-Maxwell-Poisson distribution
with parameters \mu and \sigma.
Usage
dCOMPO(x, mu, sigma, log = FALSE)
pCOMPO(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qCOMPO(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rCOMPO(n, mu, sigma)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
p |
vector of probabilities. |
n |
number of random values to return. |
Details
The COMPO distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \frac{\mu^x}{(x!)^{\sigma} Z(\mu, \sigma)}
with \mu > 0, \sigma \geq 0 and
Z(\mu, \sigma)=\sum_{j=0}^{\infty} \frac{\mu^j}{(j!)^\sigma}.
The proposed functions here are based on the functions from the COMPoissonReg package.
Value
dCOMPO gives the density, pCOMPO gives the distribution
function, qCOMPO gives the quantile function, rCOMPO
generates random deviates.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., & Boatwright, P. (2005). A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution. Journal of the Royal Statistical Society Series C: Applied Statistics, 54(1), 127-142.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 20
probs1 <- dCOMPO(x=0:x_max, mu=2, sigma=0.5)
probs2 <- dCOMPO(x=0:x_max, mu=8, sigma=1.0)
probs3 <- dCOMPO(x=0:x_max, mu=15, sigma=1.5)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for COMPO",
ylim=c(0, 0.30))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=2, sigma=0.5",
"mu=8, sigma=1.0",
"mu=15, sigma=1.5"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 20
cumulative_probs1 <- pCOMPO(q=0:x_max, mu=2, sigma=0.5)
cumulative_probs2 <- pCOMPO(q=0:x_max, mu=8, sigma=1.0)
cumulative_probs3 <- pCOMPO(q=0:x_max, mu=15, sigma=1.5)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for COMPO",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=2, sigma=0.5",
"mu=8, sigma=1.0",
"mu=15, sigma=1.5"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 50
probs1 <- dCOMPO(x=0:x_max, mu=5, sigma=0.5)
names(probs1) <- 0:x_max
x <- rCOMPO(n=1000, mu=5, sigma=0.5)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside = TRUE, names.arg = cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 3
sigma <- 1.5
p <- seq(from=0.01, to=0.99, by=0.01)
qxx <- qCOMPO(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of COMPO(mu = 3, sigma = 1.5)")
The COMPO2 distribution (with mu as mean)
Description
These functions define the density, distribution function, quantile
function and random generation for the
Comway-Maxwell-Poisson distribution
with parameters \mu and \sigma.
This parameterization was
proposed by Ribeiro et al. (2020) and the main
characteristic is that E(X)=\mu.
Usage
dCOMPO2(x, mu, sigma, log = FALSE)
pCOMPO2(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qCOMPO2(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rCOMPO2(n, mu, sigma)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
p |
vector of probabilities. |
n |
number of random values to return. |
Details
The COMPO2 distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \left(\mu + \frac{\exp(\sigma)-1}{2 \exp(\sigma)} \right)^{x \exp(\sigma)} \frac{(x!)^{\exp(\sigma)}}{Z(\mu, \sigma)}
with \mu > 0, \sigma \in \Re and
Z(\mu, \sigma)=\sum_{j=0}^{\infty} \frac{\mu^j}{(j!)^\sigma}.
The proposed functions here are based on the functions from the COMPoissonReg package.
Value
dCOMPO2 gives the density, pCOMPO2 gives the distribution
function, qCOMPO2 gives the quantile function, rCOMPO2
generates random deviates.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Ribeiro Jr, Eduardo E., et al. "Reparametrization of COM–Poisson regression models with applications in the analysis of experimental data." Statistical Modelling 20.5 (2020): 443-466.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 20
probs1 <- dCOMPO2(x=0:x_max, mu=2, sigma=-0.7)
probs2 <- dCOMPO2(x=0:x_max, mu=8, sigma=0)
probs3 <- dCOMPO2(x=0:x_max, mu=15, sigma=0.7)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for COMPO2",
ylim=c(0, 0.30))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=2, sigma=-0.7",
"mu=8, sigma=0",
"mu=15, sigma=0.7"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 20
cumulative_probs1 <- pCOMPO2(q=0:x_max, mu=2, sigma=-0.7)
cumulative_probs2 <- pCOMPO2(q=0:x_max, mu=8, sigma=0)
cumulative_probs3 <- pCOMPO2(q=0:x_max, mu=15, sigma=0.7)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for COMPO2",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=2, sigma=-0.7",
"mu=8, sigma=0",
"mu=15, sigma=0.7"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 15
probs1 <- dCOMPO2(x=0:x_max, mu=5, sigma=0.5)
names(probs1) <- 0:x_max
x <- rCOMPO2(n=1000, mu=5, sigma=0.5)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside = TRUE, names.arg = cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 3
sigma <- 0.15
p <- seq(from=0.01, to=0.99, by=0.01)
qxx <- qCOMPO2(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of COMPO2(mu = 3, sigma = 0.15)")
The Discrete Burr Hatke distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the Discrete Burr Hatke distribution
with parameter \mu.
Usage
dDBH(x, mu, log = FALSE)
pDBH(q, mu, lower.tail = TRUE, log.p = FALSE)
qDBH(p, mu = 1, lower.tail = TRUE, log.p = FALSE)
rDBH(n, mu = 1)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
p |
vector of probabilities. |
n |
number of random values to return |
Details
The Discrete Burr-Hatke distribution with parameters \mu has a support
0, 1, 2, ... and density given by
f(x | \mu) = (\frac{1}{x+1}-\frac{\mu}{x+2})\mu^{x}
The pmf is log-convex for all values of 0 < \mu < 1, where \frac{f(x+1;\mu)}{f(x;\mu)}
is an increasing function in x for all values of the parameter \mu.
Note: in this implementation we changed the original parameters \lambda for \mu,
we did it to implement this distribution within gamlss framework.
Value
dDBH gives the density, pDBH gives the distribution
function, qDBH gives the quantile function, rDBH
generates random deviates.
Author(s)
Valentina Hurtado Sepulveda, vhurtados@unal.edu.co
References
El-Morshedy, M., Eliwa, M. S., & Altun, E. (2020). Discrete Burr-Hatke distribution with properties, estimation methods and regression model. IEEE access, 8, 74359-74370.
See Also
DBH.
Examples
# Example 1
# Plotting the mass function for different parameter values
plot(x=0:5, y=dDBH(x=0:5, mu=0.1),
type="h", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", ylim=c(0, 1),
main="Probability mu=0.1")
plot(x=0:10, y=dDBH(x=0:10, mu=0.5),
type="h", lwd=2, col="tomato", las=1,
ylab="P(X=x)", xlab="X", ylim=c(0, 1),
main="Probability mu=0.5")
plot(x=0:15, y=dDBH(x=0:15, mu=0.9),
type="h", lwd=2, col="green4", las=1,
ylab="P(X=x)", xlab="X", ylim=c(0, 1),
main="Probability mu=0.9")
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 15
cumulative_probs1 <- pDBH(q=0:x_max, mu=0.1)
cumulative_probs2 <- pDBH(q=0:x_max, mu=0.5)
cumulative_probs3 <- pDBH(q=0:x_max, mu=0.9)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for Burr-Hatke",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=0.1",
"mu=0.5",
"mu=0.9"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
mu <- 0.4
x_max <- 10
probs1 <- dDBH(x=0:x_max, mu=mu)
names(probs1) <- 0:x_max
x <- rDBH(n=1000, mu=mu)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside = TRUE, names.arg = cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 0.97
p <- seq(from=0, to=1, by = 0.01)
qxx <- qDBH(p, mu, lower.tail = TRUE, log.p = FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of BH(mu=0.97)")
Discrete generalized exponential distribution - a second type
Description
These functions define the density, distribution function, quantile
function and random generation for the Discrete generalized exponential distribution
a second type with parameters \mu and \sigma.
Usage
dDGEII(x, mu = 0.5, sigma = 1.5, log = FALSE)
pDGEII(q, mu = 0.5, sigma = 1.5, lower.tail = TRUE, log.p = FALSE)
rDGEII(n, mu = 0.5, sigma = 1.5)
qDGEII(p, mu = 0.5, sigma = 1.5, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The DGEII distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = (1-\mu^{x+1})^{\sigma}-(1-\mu^x)^{\sigma}
with 0 < \mu < 1 and \sigma > 0. If \sigma=1, the DGEII distribution
reduces to the geometric distribution with success probability 1-\mu.
Note: in this implementation we changed the original parameters
p to \mu and \alpha to \sigma,
we did it to implement this distribution within gamlss framework.
Value
dDGEII gives the density, pDGEII gives the distribution
function, qDGEII gives the quantile function, rDGEII
generates random deviates.
Author(s)
Valentina Hurtado Sepulveda, vhurtados@unal.edu.co
References
Nekoukhou, V., Alamatsaz, M. H., & Bidram, H. (2013). Discrete generalized exponential distribution of a second type. Statistics, 47(4), 876-887.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 40
probs1 <- dDGEII(x=0:x_max, mu=0.1, sigma=5)
probs2 <- dDGEII(x=0:x_max, mu=0.5, sigma=5)
probs3 <- dDGEII(x=0:x_max, mu=0.9, sigma=5)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for DGEII",
ylim=c(0, 0.60))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=0.1, sigma=5",
"mu=0.5, sigma=5",
"mu=0.9, sigma=5"))
# Example 2
# Checking if the cumulative curves converge to 1
#plot1
x_max <- 10
plot_discrete_cdf(x=0:x_max,
fx=dDGEII(x=0:x_max, mu=0.3, sigma=15),
col="dodgerblue",
main="CDF for DGEII",
lwd=3)
legend("bottomright", legend="mu=0.3, sigma=15",
col="dodgerblue", lty=1, lwd=2, cex=0.8)
#plot2
plot_discrete_cdf(x=0:x_max,
fx=dDGEII(x=0:x_max, mu=0.5, sigma=30),
col="tomato",
main="CDF for DGEII",
lwd=3)
legend("bottomright", legend="mu=0.5, sigma=30",
col="tomato", lty=1, lwd=2, cex=0.8)
#plot3
plot_discrete_cdf(x=0:x_max,
fx=dDGEII(x=0:x_max, mu=0.5, sigma=50),
col="green4",
main="CDF for DGEII",
lwd=3)
legend("bottomright", legend="mu=0.5, sigma=50",
col="green4", lty=1, lwd=2, cex=0.8)
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 15
probs1 <- dDGEII(x=0:x_max, mu=0.5, sigma=5)
names(probs1) <- 0:x_max
x <- rDGEII(n=1000, mu=0.5, sigma=5)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 0.5
sigma <- 5
p <- seq(from=0, to=1, by=0.01)
qxx <- qDGEII(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of DDGEII(mu=0.5, sigma=5)")
The discrete Inverted Kumaraswamy distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the discrete Inverted Kumaraswamy, DIKUM(), distribution
with parameters \mu and \sigma.
Usage
dDIKUM(x, mu = 1, sigma = 5, log = FALSE)
pDIKUM(q, mu = 1, sigma = 5, lower.tail = TRUE, log.p = FALSE)
rDIKUM(n, mu = 1, sigma = 5)
qDIKUM(p, mu = 1, sigma = 5, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The discrete Inverted Kumaraswamy distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and density given by
f(x | \mu, \sigma) = (1-(2+x)^{-\mu})^{\sigma}-(1-(1+x)^{-\mu})^{\sigma}
with \mu > 0 and \sigma > 0.
Note: in this implementation we changed the original parameters \alpha and \beta
for \mu and \sigma respectively, we did it to implement this distribution within gamlss framework.
Value
dDIKUM gives the density, pDIKUM gives the distribution
function, qDIKUM gives the quantile function, rDIKUM
generates random deviates.
Author(s)
Daniel Felipe Villa Rengifo, dvilla@unal.edu.co
References
El-Helbawy, A. A., Hegazy, M. A., Al-Dayian, G. R., & Abd EL-Kader, R. E. (2022). A discrete analog of the inverted Kumaraswamy distribution: properties and estimation with application to COVID-19 data. Pakistan Journal of Statistics and Operation Research, 18(1), 297-328.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 30
probs1 <- dDIKUM(x=0:x_max, mu=1, sigma=5)
probs2 <- dDIKUM(x=0:x_max, mu=1, sigma=20)
probs3 <- dDIKUM(x=0:x_max, mu=1, sigma=50)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for Inverted Kumaraswamy Distribution",
ylim=c(0, 0.12))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=1, sigma=5",
"mu=1, sigma=20",
"mu=1, sigma=50"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 500
cumulative_probs1 <- pDIKUM(q=0:x_max, mu=1, sigma=5)
cumulative_probs2 <- pDIKUM(q=0:x_max, mu=1, sigma=20)
cumulative_probs3 <- pDIKUM(q=0:x_max, mu=1, sigma=50)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for Inverted Kumaraswamy Distribution",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=1, sigma=5",
"mu=1, sigma=20",
"mu=1, sigma=50"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 20
probs1 <- dDIKUM(x=0:x_max, mu=3, sigma=20)
names(probs1) <- 0:x_max
x <- rDIKUM(n=1000, mu=3, sigma=20)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
nombres <- cn
mp <- barplot(height, beside = TRUE, names.arg = nombres,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 1
sigma <- 5
p <- seq(from=0.01, to=0.99, by=0.1)
qxx <- qDIKUM(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of HP(mu = sigma = 3)")
The Discrete Lindley distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the Discrete Lindley distribution
with parameter \mu.
Usage
dDLD(x, mu, log = FALSE)
pDLD(q, mu, lower.tail = TRUE, log.p = FALSE)
qDLD(p, mu, lower.tail = TRUE, log.p = FALSE)
rDLD(n, mu = 0.5)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of positive values of this parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
p |
vector of probabilities. |
n |
number of random values to return. |
Details
The Discrete Lindley distribution with parameters \mu has a support
0, 1, 2, ... and density given by
f(x | \mu) = \frac{e^{-\mu x}}{1 + \mu} \left[ \mu(1 - 2e^{-\mu}) + (1- e^{-\mu})(1+\mu x)\right]
Note: in this implementation we changed the original parameters \theta for \mu,
we did it to implement this distribution within gamlss framework.
Value
dDLD gives the density, pDLD gives the distribution
function, qDLD gives the quantile function, rDLD
generates random deviates.
Author(s)
Yojan Andrés Alcaraz Pérez, yalcaraz@unal.edu.co
References
Bakouch, H. S., Jazi, M. A., & Nadarajah, S. (2014). A new discrete distribution. Statistics, 48(1), 200-240.
See Also
DLD.
Examples
# Example 1
# Plotting the mass function for different parameter values
plot(x=0:25, y=dDLD(x=0:25, mu=0.2),
type="h", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", ylim=c(0, 0.1),
main="Probability mu=0.2")
plot(x=0:15, y=dDLD(x=0:15, mu=0.5),
type="h", lwd=2, col="tomato", las=1,
ylab="P(X=x)", xlab="X", ylim=c(0, 0.25),
main="Probability mu=0.5")
plot(x=0:8, y=dDLD(x=0:8, mu=1),
type="h", lwd=2, col="green4", las=1,
ylab="P(X=x)", xlab="X", ylim=c(0, 0.5),
main="Probability mu=1")
plot(x=0:5, y=dDLD(x=0:5, mu=2),
type="h", lwd=2, col="red", las=1,
ylab="P(X=x)", xlab="X", ylim=c(0, 1),
main="Probability mu=2")
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 10
cumulative_probs1 <- pDLD(q=0:x_max, mu=0.2)
cumulative_probs2 <- pDLD(q=0:x_max, mu=0.5)
cumulative_probs3 <- pDLD(q=0:x_max, mu=1)
cumulative_probs4 <- pDLD(q=0:x_max, mu=2)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for Lindley",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
points(x=0:x_max, y=cumulative_probs4, type="o", col="magenta")
legend("bottomright",
col=c("dodgerblue", "tomato", "green4", "magenta"), lwd=3,
legend=c("mu=0.2",
"mu=0.5",
"mu=1",
"mu=2"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
mu <- 0.6
x <- rDLD(n = 1000, mu = mu)
x_max <- max(x)
probs1 <- dDLD(x = 0:x_max, mu = mu)
names(probs1) <- 0:x_max
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside = TRUE, names.arg = cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 0.9
p <- seq(from=0, to=1, by=0.01)
qxx <- qDLD(p, mu, lower.tail = TRUE, log.p = FALSE)
plot(p, qxx, type="S", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of DL(mu=0.9)")
The DMOLBE distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the Discrete Marshall–Olkin Length Biased
Exponential DMOLBE distribution
with parameters \mu and \sigma.
Usage
dDMOLBE(x, mu = 1, sigma = 1, log = FALSE)
pDMOLBE(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rDMOLBE(n, mu = 1, sigma = 1)
qDMOLBE(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The DMOLBE distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \frac{\sigma ((1+x/\mu)\exp(-x/\mu)-(1+(x+1)/\mu)\exp(-(x+1)/\mu))}{(1-(1-\sigma)(1+x/\mu)\exp(-x/\mu)) ((1-(1-\sigma)(1+(x+1)/\mu)\exp(-(x+1)/\mu))}
with \mu > 0 and \sigma > 0
Value
dDMOLBE gives the density, pDMOLBE gives the distribution
function, qDMOLBE gives the quantile function, rDMOLBE
generates random deviates.
Author(s)
Olga Usuga, olga.usuga@udea.edu.co
References
Aljohani, H. M., Ahsan-ul-Haq, M., Zafar, J., Almetwally, E. M., Alghamdi, A. S., Hussam, E., & Muse, A. H. (2023). Analysis of Covid-19 data using discrete Marshall–Olkinin length biased exponential: Bayesian and frequentist approach. Scientific Reports, 13(1), 12243.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 20
probs1 <- dDMOLBE(x=0:x_max, mu=0.5, sigma=0.5)
probs2 <- dDMOLBE(x=0:x_max, mu=5, sigma=0.5)
probs3 <- dDMOLBE(x=0:x_max, mu=1, sigma=2)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for DMOLBE",
ylim=c(0, 0.80))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=0.5, sigma=0.5",
"mu=5, sigma=0.5",
"mu=1, sigma=2"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 20
cumulative_probs1 <- pDMOLBE(q=0:x_max, mu=0.5, sigma=0.5)
cumulative_probs2 <- pDMOLBE(q=0:x_max, mu=5, sigma=0.5)
cumulative_probs3 <- pDMOLBE(q=0:x_max, mu=1, sigma=2)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for DMOLBE",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=0.5, sigma=0.5",
"mu=5, sigma=0.5",
"mu=1, sigma=2"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 50
probs1 <- dDMOLBE(x=0:x_max, mu=5, sigma=0.5)
names(probs1) <- 0:x_max
x <- rDMOLBE(n=1000, mu=5, sigma=0.5)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside = TRUE, names.arg = cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 3
sigma <-3
p <- seq(from=0, to=1, by=0.01)
qxx <- qDMOLBE(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of DMOLBE(mu = 3, sigma = 3)")
The Discrete Perks distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the Discrete Perks, DPERKS(),
distribution
with parameters \mu and \sigma.
Usage
dDPERKS(x, mu = 0.5, sigma = 0.5, log = FALSE)
pDPERKS(q, mu = 0.5, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rDPERKS(n, mu = 0.5, sigma = 0.5)
qDPERKS(p, mu = 0.5, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The discrete Perks distribution with parameters \mu > 0 and \sigma > 0
has a support 0, 1, 2, ... and density given by
f(x | \mu, \sigma) = \frac{\mu(1+\mu)(e^\sigma-1)e^{\sigma x}}{(1+\mu e^{\sigma x})(1+\mu e^{\sigma(x+1)})}
Note: in this implementation we changed the original parameters
\lambda for \mu and \beta for \sigma,
we did it to implement this distribution within gamlss framework.
Value
dDPERKS gives the density, pDPERKS gives the distribution
function, qDPERKS gives the quantile function, rDPERKS
generates random deviates.
Author(s)
Veronica Seguro Varela, vseguro@unal.edu.co
References
Tyagi, A., Choudhary, N., & Singh, B. (2020). A new discrete distribution: Theory and applications to discrete failure lifetime and count data. J. Appl. Probab. Statist, 15, 117-143.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 25
probs1 <- dDPERKS(x=0:x_max, mu=0.001, sigma=0.52)
probs2 <- dDPERKS(x=0:x_max, mu=0.001, sigma=0.85)
probs3 <- dDPERKS(x=0:x_max, mu=0.001, sigma=1.5)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="green4", las=1,
ylab="P(X=x)", xlab="X", main="Probability for DPERKS",
ylim=c(0, 0.40))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="black")
legend("topright", col=c("green4", "tomato", "black"), lwd=3,
legend=c("mu=0.001, sigma=0.52 ",
"mu=0.001, sigma=0.85",
"mu=0.001, sigma=1.5"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 25
cumulative_probs1 <- pDPERKS(q=0:x_max, mu=0.001, sigma=0.52)
cumulative_probs2 <- pDPERKS(q=0:x_max, mu=0.001, sigma=0.85)
cumulative_probs3 <- pDPERKS(q=0:x_max, mu=0.001, sigma=1.5)
plot(x=0:x_max, y=cumulative_probs1, col="green4",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for DPERKS",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="black")
legend("bottomright", col=c("green4", "tomato", "black"), lwd=3,
legend=c("mu=0.001, sigma=0.52 ",
"mu=0.001, sigma=0.85",
"mu=0.001, sigma=1.5"))
# Example 3
# Comparing the random generator output with the theoretical probabilities
x_max <- 20
mu <- 2.5
sigma <- 0.4
probs1 <- dDPERKS(x=0:x_max, mu=mu, sigma=sigma)
names(probs1) <- 0:x_max
x <- rDPERKS(n=10000, mu=mu, sigma=sigma)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
nombres <- cn
mp <- barplot(height, beside = TRUE, names.arg = nombres,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 0.2
sigma <- 0.2
p <- seq(from=0, to=1, by=0.01)
qxx <- qDPERKS(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of DPERKS(mu = sigma = 0.2)")
The DsPA distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the
discrete power-Ailamujia distribution
with parameters \mu and \sigma.
Usage
dDsPA(x, mu, sigma, log = FALSE)
pDsPA(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qDsPA(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rDsPA(n, mu, sigma)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
p |
vector of probabilities. |
n |
number of random values to return. |
Details
The DsPA distribution with parameters \mu and \sigma has a support
0, 1, 2, ...
Note:in this implementation we changed the original parameters
\beta and \lambda for \mu and \sigma respectively,
we did it to implement this distribution within gamlss framework.
Value
dDsPA gives the density, pDsPA gives the distribution
function, qDsPA gives the quantile function.
Author(s)
Maria Camila Mena Romana, mamenar@unal.edu.co
References
Alghamdi, A. S., Ahsan-ul-Haq, M., Babar, A., Aljohani, H. M., Afify, A. Z., & Cell, Q. E. (2022). The discrete power-Ailamujia distribution: properties, inference, and applications. AIMS Math, 7(5), 8344-8360.
See Also
DsPA.
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 30
probs1 <- dDsPA(x=0:x_max, mu=1.2, sigma=0.5)
probs2 <- dDsPA(x=0:x_max, mu=1.2, sigma=0.7)
probs3 <- dDsPA(x=0:x_max, mu=1.2, sigma=0.9)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for DsPA",
ylim=c(0, 0.40))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=1.2, sigma=0.5",
"mu=1.2, sigma=0.7",
"mu=1.2, sigma=0.9"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 15
cumulative_probs1 <- pDsPA(q=0:x_max, mu=1.2, sigma=0.5)
cumulative_probs2 <- pDsPA(q=0:x_max, mu=1.2, sigma=0.7)
cumulative_probs3 <- pDsPA(q=0:x_max, mu=1.2, sigma=0.9)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for DsPA",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=1.2, sigma=0.5",
"mu=1.2, sigma=0.7",
"mu=1.2, sigma=0.9"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 50
probs1 <- dDsPA(x=0:x_max, mu=1.2, sigma=0.9)
names(probs1) <- 0:x_max
x <- rDsPA(n=1000, mu=1.2, sigma=0.9)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 1.2
sigma <- 0.9
p <- seq(from=0, to=1, by=0.01)
qxx <- qDsPA(p=p, mu=mu, sigma=sigma,
lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of DsPA(mu=1.2, sigma=0.9)")
The GGEO distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the Generalized Geometric distribution
with parameters \mu and \sigma.
Usage
dGGEO(x, mu = 0.5, sigma = 1, log = FALSE)
pGGEO(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rGGEO(n, mu = 0.5, sigma = 1)
qGGEO(p, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The GGEO distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and mass function given by
f(x | \mu, \sigma) = \frac{\sigma \mu^x (1-\mu)}{(1-(1-\sigma) \mu^{x+1})(1-(1-\sigma) \mu^{x})}
with 0 < \mu < 1 and \sigma > 0. If \sigma=1, the GGEO distribution
reduces to the geometric distribution with success probability 1-\mu.
Note: in this implementation we changed the original parameters
\theta for \mu and \alpha for \sigma,
we did it to implement this distribution within gamlss framework.
Value
dGGEO gives the density, pGGEO gives the distribution
function, qGGEO gives the quantile function, rGGEO
generates random deviates.
Author(s)
Valentina Hurtado Sepulveda, vhurtados@unal.edu.co
References
Gómez-Déniz, E. (2010). Another generalization of the geometric distribution. Test, 19, 399-415.
See Also
GGEO.
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 80
probs1 <- dGGEO(x=0:x_max, mu=0.5, sigma=10)
probs2 <- dGGEO(x=0:x_max, mu=0.7, sigma=30)
probs3 <- dGGEO(x=0:x_max, mu=0.9, sigma=50)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for GGEO",
ylim=c(0, 0.20))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=0.5, sigma=10",
"mu=0.7, sigma=30",
"mu=0.9, sigma=50"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 10
plot_discrete_cdf(x=0:x_max,
fx=dGGEO(x=0:x_max, mu=0.3, sigma=15),
col="dodgerblue",
main="CDF for GGEO",
lwd= 3)
legend("bottomright", legend="mu=0.3, sigma=15", col="dodgerblue",
lty=1, lwd=2, cex=0.8)
plot_discrete_cdf(x=0:x_max,
fx=dGGEO(x=0:x_max, mu=0.5, sigma=30),
col="tomato",
main="CDF for GGEO",
lwd=3)
legend("bottomright", legend="mu=0.5, sigma=30",
col="tomato", lty=1, lwd=2, cex=0.8)
plot_discrete_cdf(x=0:x_max,
fx=dGGEO(x=0:x_max, mu=0.5, sigma=50),
col="green4",
main="CDF for GGEO",
lwd=3)
legend("bottomright", legend="mu=0.5, sigma=50",
col="green4", lty=1, lwd=2, cex=0.8)
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 15
probs1 <- dGGEO(x=0:x_max, mu=0.5, sigma=5)
names(probs1) <- 0:x_max
x <- rGGEO(n=1000, mu=0.5, sigma=5)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
col=c("dodgerblue3", "firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 0.5
sigma <- 5
p <- seq(from=0, to=1, by=0.01)
qxx <- qGGEO(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of GGEO(mu=0.5, sigma=0.5)")
The hyper-Poisson distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the hyper-Poisson, HYPERPO(), distribution
with parameters \mu and \sigma.
Usage
dHYPERPO(x, mu = 1, sigma = 1, log = FALSE)
pHYPERPO(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rHYPERPO(n, mu = 1, sigma = 1)
qHYPERPO(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The hyper-Poisson distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and density given by
f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}
where the function _1F_1(a;c;z) is defined as
_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}
and (a)_r = \frac{\gamma(a+r)}{\gamma(a)} for a>0 and r positive integer.
Note: in this implementation we changed the original parameters \lambda and \gamma
for \mu and \sigma respectively, we did it to implement this distribution within gamlss framework.
Value
dHYPERPO gives the density, pHYPERPO gives the distribution
function, qHYPERPO gives the quantile function, rHYPERPO
generates random deviates.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Sáez-Castillo, A. J., & Conde-Sánchez, A. (2013). A hyper-Poisson regression model for overdispersed and underdispersed count data. Computational Statistics & Data Analysis, 61, 148-157.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 30
probs1 <- dHYPERPO(x=0:x_max, mu=5, sigma=0.1)
probs2 <- dHYPERPO(x=0:x_max, mu=5, sigma=1.0)
probs3 <- dHYPERPO(x=0:x_max, mu=5, sigma=1.8)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for hyper-Poisson",
ylim=c(0, 0.20))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=5, sigma=0.1",
"mu=5, sigma=1.0",
"mu=5, sigma=1.8"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 15
cumulative_probs1 <- pHYPERPO(q=0:x_max, mu=5, sigma=0.1)
cumulative_probs2 <- pHYPERPO(q=0:x_max, mu=5, sigma=1.0)
cumulative_probs3 <- pHYPERPO(q=0:x_max, mu=5, sigma=1.8)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for hyper-Poisson",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=5, sigma=0.1",
"mu=5, sigma=1.0",
"mu=5, sigma=1.8"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 15
probs1 <- dHYPERPO(x=0:x_max, mu=3, sigma=1.1)
names(probs1) <- 0:x_max
x <- rHYPERPO(n=1000, mu=3, sigma=1.1)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
nombres <- cn
mp <- barplot(height, beside = TRUE, names.arg = nombres,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 3
sigma <-3
p <- seq(from=0, to=1, by=0.01)
qxx <- qHYPERPO(p=p, mu=mu, sigma=sigma,
lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of HP(mu=3, sigma=3)")
The hyper-Poisson distribution (with mu as mean)
Description
These functions define the density, distribution function, quantile
function and random generation for the hyper-Poisson in
the second parameterization with parameters \mu (as mean) and
\sigma as the dispersion parameter.
Usage
dHYPERPO2(x, mu = 1, sigma = 1, log = FALSE)
pHYPERPO2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rHYPERPO2(n, mu = 1, sigma = 1)
qHYPERPO2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of positive values of this parameter. |
sigma |
vector of positive values of this parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return |
p |
vector of probabilities. |
Details
The hyper-Poisson distribution with parameters \mu and \sigma
has a support 0, 1, 2, ...
Note: in this implementation the parameter \mu is the mean
of the distribution and \sigma corresponds to
the dispersion parameter. If you fit a model with this parameterization,
the time will increase because an internal procedure to convert \mu
to \lambda parameter.
Value
dHYPERPO2 gives the density, pHYPERPO2 gives the distribution
function, qHYPERPO2 gives the quantile function, rHYPERPO2
generates random deviates.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Sáez-Castillo, A. J., & Conde-Sánchez, A. (2013). A hyper-Poisson regression model for overdispersed and underdispersed count data. Computational Statistics & Data Analysis, 61, 148-157.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 30
probs1 <- dHYPERPO2(x=0:x_max, sigma=0.01, mu=3)
probs2 <- dHYPERPO2(x=0:x_max, sigma=0.50, mu=5)
probs3 <- dHYPERPO2(x=0:x_max, sigma=1.00, mu=7)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for hyper-Poisson",
ylim=c(0, 0.30))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("sigma=0.01, mu=3",
"sigma=0.50, mu=5",
"sigma=1.00, mu=7"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 15
cumulative_probs1 <- pHYPERPO2(q=0:x_max, mu=1, sigma=1.5)
cumulative_probs2 <- pHYPERPO2(q=0:x_max, mu=3, sigma=1.5)
cumulative_probs3 <- pHYPERPO2(q=0:x_max, mu=5, sigma=1.5)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for hyper-Poisson",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("sigma=1.5, mu=1",
"sigma=1.5, mu=3",
"sigma=1.5, mu=5"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 15
probs1 <- dHYPERPO2(x=0:x_max, mu=3, sigma=1.1)
names(probs1) <- 0:x_max
x <- rHYPERPO2(n=1000, mu=3, sigma=1.1)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
nombres <- cn
mp <- barplot(height, beside = TRUE, names.arg = nombres,
col=c('dodgerblue3','firebrick3'), las=1,
xlab='X', ylab='Proportion')
legend('topright',
legend=c('Theoretical', 'Simulated'),
bty='n', lwd=3,
col=c('dodgerblue3','firebrick3'), lty=1)
# Example 4
# Checking the quantile function
mu <- 3
sigma <-3
p <- seq(from=0, to=1, by=0.01)
qxx <- qHYPERPO2(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of HP2(mu = sigma = 3)")
New Poisson-generalised Lindley distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the Poisson-generalised Lindley (NPGL)
distribution
with parameters \mu and \sigma.
Usage
dNPGL(x, mu = 0.1, sigma = 2, log = FALSE)
pNPGL(q, mu = 0.1, sigma = 2, lower.tail = TRUE, log.p = FALSE)
rNPGL(n, mu = 0.1, sigma = 2)
qNPGL(p, mu = 0.1, sigma = 2, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The Poisson-generalised Lindley distribution with parameters \mu and \sigma
has support x = 0, 1, 2, \ldots and probability mass function given by
f(x \mid \mu, \sigma)=\frac{\mu^2+\frac{\mu^{\sigma}(\mu+1)^{1-\sigma}\Gamma(x+\sigma)}{\Gamma(\sigma)\Gamma(x+1)}}{(\mu+1)^{x+2}}
with \mu > 0 and \sigma > 0.
This distribution is useful for modeling over-dispersed count data.
Note: in this implementation we changed the original parameters \theta and \alpha
for \mu and \sigma respectively, we did it to implement this distribution within gamlss framework.
Value
dNPGL gives the density, pNPGL gives the distribution
function, qNPGL gives the quantile function, rNPGL
generates random deviates.
Author(s)
Tomas Mesa, tomas.mesaz@udea.edu.co
References
Altun, E. A new two-parameter discrete poisson-generalized Lindley distribution with properties and applications to healthcare data sets. Comput Stat 36, 2841–2861 (2021). https://doi.org/10.1007/s00180-021-01097-0
See Also
NPGL.
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 50
probs1 <- dNPGL(x=0:x_max, mu = 0.1, sigma = 2)
probs2 <- dNPGL(x=0:x_max, mu = 0.5, sigma = 5)
probs3 <- dNPGL(x=0:x_max, mu = 0.2, sigma = 6)
probs4 <- dNPGL(x=0:x_max, mu = 20, sigma = 2)
plot(x=0:x_max, y=probs1, type="h", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for dNPGL",
ylim=c(0, 0.035))
legend("topright", legend="mu=0.1, sigma=2")
plot(x=0:x_max, y=probs2, type="h", lwd=2, col="tomato", las=1,
ylab="P(X=x)", xlab="X", main="Probability for dNPGL",
ylim=c(0, 0.1))
legend("topright", legend="mu=0.5, sigma=5")
plot(x=0:x_max, y=probs3, type="h", lwd=2, col="green4", las=1,
ylab="P(X=x)", xlab="X", main="Probability for dNPGL",
ylim=c(0, 0.03))
legend("topright", legend="mu=0.2, sigma=6")
plot(x=0:x_max, y=probs4, type="h", lwd=2, col="magenta", las=1,
ylab="P(X=x)", xlab="X", main="Probability for dNPGL",
ylim=c(0, 1))
legend("topright", legend="mu=20, sigma=2")
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 100
cumulative_probs1 <- pNPGL(q=0:x_max, mu = 0.1, sigma = 2)
cumulative_probs2 <- pNPGL(q=0:x_max, mu = 0.5, sigma = 5)
cumulative_probs3 <- pNPGL(q=0:x_max, mu = 0.2, sigma = 6)
cumulative_probs4 <- pNPGL(q=0:x_max, mu = 20, sigma = 2)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative probability for NPGL",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
points(x=0:x_max, y=cumulative_probs4, type="o", col="magenta")
legend("bottomright", col=c("dodgerblue", "tomato", "green4", "magenta"), lwd=3,
legend=c("mu=0.1, sigma=2",
"mu=0.5, sigma=5",
"mu=0.2, sigma=6",
"mu=20, sigma=2"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 100
mu <- 0.1
sigma <- 2
probs1 <- dNPGL(x=0:x_max, mu=mu, sigma=sigma)
names(probs1) <- 0:x_max
x <- rNPGL(n=10000, mu=mu, sigma=sigma)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
nombres <- cn
mp <- barplot(height, beside = TRUE, names.arg = nombres,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 0.1
sigma <- 2
p <- seq(from=0, to=1, by=0.01)
qxx <- qNPGL(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of NPGL(mu=0.1, sigma=2)")
New Poisson-generalised Lindley distribution (with mu as mean)
Description
These functions define the density, distribution function, quantile
function and random generation for the Poisson-generalised Lindley in second
parametrization with parameters \mu (as mean) and \sigma.
Usage
dNPGL2(x, mu = 2, sigma = 2, log = FALSE)
pNPGL2(q, mu = 2, sigma = 2, lower.tail = TRUE, log.p = FALSE)
rNPGL2(n, mu = 2, sigma = 2)
qNPGL2(p, mu = 2, sigma = 2, lower.tail = TRUE, log.p = FALSE)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
n |
number of random values to return. |
p |
vector of probabilities. |
Details
The Poisson-generalised Lindley distribution with parameters \mu and \sigma
has support x = 0, 1, 2, \ldots and probability mass function given by
Note: in this implementation the parameter \mu is the mean
of the distribution and \sigma (equivalent to \alpha
in the original NPGL parametrization).
Value
dNPGL2 gives the density, pNPGL2 gives the distribution
function, qNPGL2 gives the quantile function, rNPGL2
generates random deviates.
Author(s)
Tomás Mesa, tomas.mesaz@udea.edu.co
References
Altun, E. A new two-parameter discrete poisson-generalized Lindley distribution with properties and applications to healthcare data sets. Comput Stat 36, 2841–2861 (2021). https://doi.org/10.1007/s00180-021-01097-0
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
# Original parameters and their corresponding means
casos <- data.frame(
mu_orig = c(0.1, 0.5, 0.2, 20),
sigma = c(2, 5, 6, 2)
)
casos$mu_mean <- (casos$sigma + casos$mu_orig) /
(casos$mu_orig * (1 + casos$mu_orig))
x_max <- 50
probs1 <- dNPGL2(x=0:x_max, mu=casos$mu_mean[1], sigma=casos$sigma[1])
probs2 <- dNPGL2(x=0:x_max, mu=casos$mu_mean[2], sigma=casos$sigma[2])
probs3 <- dNPGL2(x=0:x_max, mu=casos$mu_mean[3], sigma=casos$sigma[3])
probs4 <- dNPGL2(x=0:x_max, mu=casos$mu_mean[4], sigma=casos$sigma[4])
plot(x = 0:x_max, y = probs1, type = "h", lwd = 2, col = "dodgerblue", las = 1,
ylab = "P(X=x)", xlab = "X", main = "Probability for dNPGL2",
ylim = c(0, 0.035))
legend("topright", legend = paste0("mu=", round(casos$mu_mean[1], 2),
", sigma=", casos$sigma[1]))
plot(x = 0:x_max, y = probs2, type = "h", lwd = 2, col = "tomato", las = 1,
ylab = "P(X=x)", xlab = "X", main = "Probability for dNPGL2",
ylim = c(0, 0.1))
legend("topright", legend = paste0("mu=", round(casos$mu_mean[2], 2),
", sigma=", casos$sigma[2]))
plot(x = 0:x_max, y = probs3, type = "h", lwd = 2, col = "green4", las = 1,
ylab = "P(X=x)", xlab = "X", main = "Probability for dNPGL2",
ylim = c(0, 0.03))
legend("topright", legend = paste0("mu=", round(casos$mu_mean[3], 2),
", sigma=", casos$sigma[3]))
plot(x = 0:x_max, y = probs4, type = "h", lwd = 2, col = "magenta", las = 1,
ylab = "P(X=x)", xlab = "X", main = "Probability for dNPGL2",
ylim = c(0, 1))
legend("topright", legend = paste0("mu=", round(casos$mu_mean[4], 4),
", sigma=", casos$sigma[4]))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 100
cumulative_probs1 <- pNPGL2(q = 0:x_max, mu = casos$mu_mean[1], sigma = casos$sigma[1])
cumulative_probs2 <- pNPGL2(q = 0:x_max, mu = casos$mu_mean[2], sigma = casos$sigma[2])
cumulative_probs3 <- pNPGL2(q = 0:x_max, mu = casos$mu_mean[3], sigma = casos$sigma[3])
cumulative_probs4 <- pNPGL2(q = 0:x_max, mu = casos$mu_mean[4], sigma = casos$sigma[4])
plot(x = 0:x_max, y = cumulative_probs1, col = "dodgerblue",
type = "o", las = 1, ylim = c(0, 1),
main = "Cumulative probability for NPGL2",
xlab = "X", ylab = "Probability")
points(x = 0:x_max, y = cumulative_probs2, type = "o", col = "tomato")
points(x = 0:x_max, y = cumulative_probs3, type = "o", col = "green4")
points(x = 0:x_max, y = cumulative_probs4, type = "o", col = "magenta")
legend("bottomright",
col = c("dodgerblue", "tomato", "green4", "magenta"), lwd = 3,
legend = c(
paste0("mu=", round(casos$mu_mean[1], 2), ", sigma=", casos$sigma[1]),
paste0("mu=", round(casos$mu_mean[2], 2), ", sigma=", casos$sigma[2]),
paste0("mu=", round(casos$mu_mean[3], 2), ", sigma=", casos$sigma[3]),
paste0("mu=", round(casos$mu_mean[4], 4), ", sigma=", casos$sigma[4])
))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 60
mu <- 5
sigma <- 2
probs1 <- dNPGL2(x = 0:x_max, mu = mu, sigma = sigma)
names(probs1) <- 0:x_max
set.seed(123)
x <- rNPGL2(n = 10000, mu = mu, sigma = sigma)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside = TRUE, names.arg = cn,
col = c("dodgerblue3", "firebrick3"), las = 1,
xlab = "X", ylab = "Proportion",
main = paste0("NPGL2(mu=", mu, ", sigma=", sigma, ")"))
legend("topright",
legend = c("Theoretical", "Simulated"),
bty = "n", lwd = 3,
col = c("dodgerblue3", "firebrick3"), lty = 1)
# Example 4
# Checking the quantile function
mu <- 5
sigma <- 2
p <- seq(from = 0, to = 1, by = 0.01)
qxx <- qNPGL2(p = p, mu = mu, sigma = sigma, lower.tail = TRUE, log.p = FALSE)
plot(p, qxx, type = "s", lwd = 2, col = "green3", ylab = "quantiles",
main = paste0("Quantiles of NPGL2(mu=", mu, ", sigma=", sigma, ")"))
The Discrete Poisson XLindley
Description
These functions define the density, distribution function, quantile
function and random generation for the Discrete Poisson XLindley distribution
with parameter \mu.
Usage
dPOISXL(x, mu = 0.3, log = FALSE)
pPOISXL(q, mu = 0.3, lower.tail = TRUE, log.p = FALSE)
qPOISXL(p, mu = 0.3, lower.tail = TRUE, log.p = FALSE)
rPOISXL(n, mu = 0.3)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are |
p |
vector of probabilities. |
n |
number of random values to return |
Details
The Discrete Poisson XLindley distribution with parameters \mu has a support
0, 1, 2, ... and mass function given by
f(x | \mu) = \frac{\mu^2(x+\mu^2+3(1+\mu))}{(1+\mu)^{4+x}}; with \mu>0.
Note: in this implementation we changed the original parameters \alpha for \mu,
we did it to implement this distribution within gamlss framework.
Value
dPOISXL gives the density, pPOISXL gives the distribution
function, qPOISXL gives the quantile function, rPOISXL
generates random deviates.
Author(s)
Mariana Blandon Mejia, mblandonm@unal.edu.co
References
Ahsan-ul-Haq, M., Al-Bossly, A., El-Morshedy, M., & Eliwa, M. S. (2022). Poisson XLindley distribution for count data: statistical and reliability properties with estimation techniques and inference. Computational Intelligence and neuroscience, 2022(1), 6503670.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 20
probs1 <- dPOISXL(x=0:x_max, mu=0.2)
probs2 <- dPOISXL(x=0:x_max, mu=0.5)
probs3 <- dPOISXL(x=0:x_max, mu=1.0)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for Poisson XLindley",
ylim=c(0, 0.50))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=0.2", "mu=0.5", "mu=1.0"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 20
plot_discrete_cdf(x=0:x_max,
fx=dPOISXL(x=0:x_max, mu=0.2), col="dodgerblue",
main="CDF for Poisson XLindley with mu=0.2")
plot_discrete_cdf(x=0:x_max,
fx=dPOISXL(x=0:x_max, mu=0.5), col="tomato",
main="CDF for Poisson XLindley with mu=0.5")
plot_discrete_cdf(x=0:x_max,
fx=dPOISXL(x=0:x_max, mu=1.0), col="green4",
main="CDF for Poisson XLindley with mu=1.0")
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 15
probs1 <- dPOISXL(x=0:x_max, mu=0.3)
names(probs1) <- 0:x_max
x <- rPOISXL(n=3000, mu=0.3)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside = TRUE, names.arg = cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 0.3
p <- seq(from=0, to=1, by = 0.01)
qxx <- qPOISXL(p, mu, lower.tail = TRUE, log.p = FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles for Poisson XLindley mu=0.3")
The Poisson–transmuted record type exponential distribution
Description
These functions define the density, distribution function, quantile
function and random generation for the
Poisson–transmuted record type exponential (PTRTE) distribution
with parameters \mu and \sigma.
This distribution was proposed by Erbayram and Akdogan (2025) as a new discrete distribution obtained from a mixed Poisson model, where the Poisson parameter follows a transmuted record type exponential distribution.
Usage
dPTRTE(x, mu, sigma, log = FALSE)
pPTRTE(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qPTRTE(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rPTRTE(n, mu, sigma)
Arguments
x, q |
vector of (non-negative integer) quantiles. |
mu |
vector of the mu parameter. |
sigma |
vector of the sigma parameter. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are
|
p |
vector of probabilities. |
n |
number of random values to return. |
Details
The Poisson–transmuted record type exponential distribution with
parameters \mu and \sigma has support
x = 0,1,2,\dots and probability mass function given by
f(x | \mu, \sigma) = \frac{\mu}{(1+\mu)^{x+1}} \left(\frac{\sigma \mu (1+x)}{1+\mu} - (\sigma-1) \right)
with \mu > 0 and 0 < \sigma < 1.
Value
dPTRTE gives the density,
pPTRTE gives the distribution function,
qPTRTE gives the quantile function,
rPTRTE generates random deviates.
Author(s)
Rebeca Isabel Rodriguez Gonzalez, rebeca.rodriguez@udea.edu.co
References
Erbayram, T., & Akdogan, Y. (2025). A new discrete model generated from mixed Poisson transmuted record type exponential distribution. Ricerca di Matematica, 74, 1225–1247.
See Also
Examples
# Example 1
# Plotting the mass function for different parameter values
x_max <- 20
probs1 <- dPTRTE(x=0:x_max, mu=0.5, sigma=0.5)
probs2 <- dPTRTE(x=0:x_max, mu=1, sigma=0.5)
probs3 <- dPTRTE(x=0:x_max, mu=2, sigma=0.5)
# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
ylab="P(X=x)", xlab="X", main="Probability for PTRTE",
ylim=c(0, 0.80))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=0.5, sigma=0.5",
"mu=1, sigma=0.5",
"mu=2, sigma=0.5"))
# Example 2
# Checking if the cumulative curves converge to 1
x_max <- 10
cumulative_probs1 <- pPTRTE(q=0:x_max, mu=1, sigma=0.5)
cumulative_probs2 <- pPTRTE(q=0:x_max, mu=1, sigma=0.7)
cumulative_probs3 <- pPTRTE(q=0:x_max, mu=1, sigma=0.9)
plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
type="o", las=1, ylim=c(0, 1),
main="Cumulative for PTRTE",
xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
legend=c("mu=1, sigma=0.5",
"mu=1, sigma=0.7",
"mu=1, sigma=0.9"))
# Example 3
# Comparing the random generator output with
# the theoretical probabilities
x_max <- 30
probs1 <- dPTRTE(x=0:x_max, mu=0.5, sigma=0.5)
names(probs1) <- 0:x_max
x <- rPTRTE(n=1000, mu=0.5, sigma=0.5)
probs2 <- prop.table(table(x))
cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
col=c("dodgerblue3","firebrick3"), las=1,
xlab="X", ylab="Proportion")
legend("topright",
legend=c("Theoretical", "Simulated"),
bty="n", lwd=3,
col=c("dodgerblue3","firebrick3"), lty=1)
# Example 4
# Checking the quantile function
mu <- 1
sigma <- 0.5
p <- seq(from=0, to=1, by=0.01)
qxx <- qPTRTE(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
main="Quantiles of DPTRTE(mu=1, sigma=0.5)")
Initial values for Discrete Burr Hatke
Description
This function generates initial values for the parameter mu.
Usage
estim_mu_DBH(y)
Arguments
y |
vector with the response variable. |
Value
returns a scalar with the MLE estimation.
Initial values for Discrete Lindley
Description
This function generates initial values for the parameter mu.
Usage
estim_mu_DLD(y)
Arguments
y |
vector with the response variable. |
Value
returns a scalar with the MLE estimation.
Initial values for discrete Poisson XLindley distribution
Description
This function generates initial values for the parameters.
Usage
estim_mu_POISXL(y)
Arguments
y |
vector with the response variable. |
Value
returns a scalar with the MLE estimation.
Initial values for COMPO
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_COMPO(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the moments estimations.
Initial values for DGEII
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_DGEII(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
Initial values for discrete Inverted Kumaraswamy
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_DIKUM(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
Initial values for DMOLBE
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_DMOLBE(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
Initial values for DPERKS
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_DPERKS(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
Initial values for DsPA
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_DsPA(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
Initial values for GGEO
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_GGEO(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
Initial values for hyper Poisson
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_HYPERPO(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
Initial values for hyper Poisson in second parameterization
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_HYPERPO2(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
estim_mu_sigma_NPGL
Description
This function generates initial values for NPGL distribution.
Usage
estim_mu_sigma_NPGL(y)
Arguments
y |
vector with the random sample |
Examples
y <- rNPGL(n=100, mu=0.1, sigma=2)
estim_mu_sigma_NPGL(y=y)
estim_mu_sigma_NPGL2
Description
estim_mu_sigma_NPGL2
Usage
estim_mu_sigma_NPGL2(y)
Arguments
y |
vector with the random sample |
Examples
y <- rNPGL2(n = 100, mu = 3, sigma = 2)
estim_mu_sigma_NPGL2(y = y)
estim_mu_sigma_NPGL_MM
Description
This function generates initial values for NPGL distribution using the method of moments estimators given in equations (19) and (20) of Altun (2021).
Usage
estim_mu_sigma_NPGL_MM(y)
Arguments
y |
vector with the random sample |
Examples
y <- rNPGL(n=100, mu=0.1, sigma=2)
estim_mu_sigma_NPGL_MM(y=y)
Initial values for PTRTE
Description
This function generates initial values for the parameters.
Usage
estim_mu_sigma_PTRTE(y)
Arguments
y |
vector with the response variable. |
Value
returns a vector with the MLE estimations.
grazing dataset
Description
In this experiment, the density of understorey birds at a series of sites in two areas either side of a stockproof fence were compared. Once side had limited grazing (mainly from native herbivores), and the other was heavily grazed by feral herbivores, mostly horses. Bird counts were done at the sites either side of the fence (the Before measurements). Then the herbivores were removed, and bird counts done again (the After measurements). The measurements are the total number of understorey-foraging birds observed in three 20-minute surveys of two hectare quadrats.
Usage
grazing
Format
grazing
A data frame with 62 rows and 3 variables:
- when
when the bird count was conducted
- grazed
a factor with levels Reference and Feral
- birds
the number of understorey birds
...
Author(s)
Rodrigo Matheus
Source
https://github.com/rdmatheus/bergreg
logLik function for Discrete Burr Hatke
Description
Calculates logLik for Discrete Burr Hatke distribution.
Usage
logLik_DBH(param = 0.5, x)
Arguments
param |
value for mu. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for DGEII
Description
Calculates logLik for DGEII distribution.
Usage
logLik_DGEII(transf_param = c(0, 0), x)
Arguments
transf_param |
vector with parameters in log and logit scale. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for discrete Inverted Kumaraswamy
Description
Calculates logLik for discrete Inverted Kumaraswamy distribution.
Usage
logLik_DIKUM(param = c(0, 0), x)
Arguments
param |
vector with parameters in log scale. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for Discrete Lindley distribution
Description
Calculates logLik for Discrete Lindley distribution.
Usage
logLik_DLD(param = 0.5, x)
Arguments
param |
value for mu. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for DMOLBE
Description
Calculates logLik for DMOLBE distribution.
Usage
logLik_DMOLBE(logparam = c(0, 0), x)
Arguments
logparam |
vector with parameters in log scale. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for DPERKS
Description
Calculates logLik for DPERKS distribution.
Usage
logLik_DPERKS(logparam = c(0, 0), x)
Arguments
logparam |
vector with parameters in log scale. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for DsPA
Description
Calculates logLik for DsPA distribution.
Usage
logLik_DsPA(param = c(0, 0), x)
Arguments
param |
a vector with parameters. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for GGEO
Description
Calculates logLik for GGEO distribution.
Usage
logLik_GGEO(param = c(0, 0), x)
Arguments
param |
vector with parameters in log and logit scale. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for hyper Poisson
Description
Calculates logLik for hyper Poisson distribution.
Usage
logLik_HYPERPO(logparam = c(0, 0), x)
Arguments
logparam |
vector with parameters in log scale. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik function for hyper Poisson in second parameterization
Description
Calculates logLik for hyper Poisson distribution.
Usage
logLik_HYPERPO2(logparam = c(0, 0), x)
Arguments
logparam |
vector with parameters in log scale. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
logLik_NPGL
Description
This is an auxiliar function to obtain the logLik for NPGL.
Usage
logLik_NPGL(param = c(0, 0), x)
Arguments
param |
vector with the values for mu and sigma |
x |
vector with the data |
Examples
y <- rNPGL(n=100, mu=0.1, sigma=2)
logLik_NPGL(param=c(0, 0), x=y)
logLik_NPGL2
Description
Auxiliary function to evaluate the log-likelihood of the mean-parametrized NPGL2 distribution.
Usage
logLik_NPGL2(param = c(log(3), log(2)), x)
Arguments
param |
vector with the values for mu and sigma |
x |
vector with the data |
Examples
y <- rNPGL2(n = 100, mu = 3, sigma = 2)
logLik_NPGL2(param = c(log(3), log(2)), x = y)
logLik function for Poisson XLindley distribution
Description
Calculates logLik for Poisson XLindley distribution distribution.
Usage
logLik_POISXL(param = 0, x)
Arguments
param |
parameter mu in log scale. |
x |
vector with the response variable. |
Value
returns the loglikelihood given the parameters and random sample.
Create a Link for GAMLSS families
Description
The function make.link.gamlss() is used with gamlss.family
distributions in package gamlss().Given a link, it returns a link
function, an inverse link function, the derivative dpar/deta where 'par'
is the appropriate distribution parameter and a function for checking
the domain. It differs from the usual make.link of glm() by
having extra links as the logshifto1, and the own.
For the use of the own link see the example bellow.
show.link provides a way in which the user can identify the link
functions available for each gamlss distribution. If your required link
function is not available for any of the gamlss distributions you can add it in.
Usage
make.link.gamlss(link)
Arguments
link |
character or numeric; one of |
Details
The own link function is added to allow the user greater
flexibility. In order to used the own link function for any of the
parameters of the distribution the own link should appear in the
available links for this parameter. You can check this using the function
show.link. If the own do not appear in the list you can
create a new function for the distribution in which own is added
in the list. For example the first line of the code of the binomial
distribution, BI, has change from
"mstats <- checklink("mu.link", "Binomial", substitute(mu.link), c("logit", "probit", "cloglog", "log")), in version 1.0-0 of gamlss, to
"mstats <- checklink("mu.link", "Binomial", substitute(mu.link), c("logit", "probit", "cloglog", "log", "own"))
in version 1.0-1. Given that the parameter has own as an option the user needs also
to define the following four new functions in order to used an
own link.
i) own.linkfun
ii) own.linkinv
iii) own.mu.eta and
iv) own.valideta.
An example is given below.
Only one parameter of the distribution at a time is allowed to have
its own link, (unless the same four own functions above
are suitable for more that one parameter of the distribution).
Note that from gamlss version 1.9-0 the user can introduce its own link function by define an appropriate function, (see the example below).
Value
For the make.link.gamlss a list with components
linkfun: Link function function(parameter)
linkinv: Inverse link function function(eta)
mu.eta: Derivative function(eta) dparameter/deta
valideta: function(eta) TRUE if all of eta is in the domain of linkinv.
For the show.link a list with components the available links for the distribution parameters
Note
For the links involving parameters as in logshifted and
logitshifted the parameters can be passed in the definition of the
distribution by calling the checklink function, for example in the
definition of the tau parameter in BCPE distribution the following
call is made: tstats <- checklink("tau.link",
"Box Cox Power Exponential", substitute(tau.link),
c("logshifted", "log", "identity"), par.link = c(1))
This function is the same make.link.gamlss fuction from gamlss.dist package.
Mean and variance for hyper-Poisson distribution
Description
This function calculates the mean and variance for the
hyper-Poisson distribution with parameters \mu and \sigma.
Usage
mean_var_hp(mu, sigma)
mean_var_hp2(mu, sigma)
Arguments
mu |
value of the mu parameter. |
sigma |
value of the sigma parameter. |
Details
The hyper-Poisson distribution with parameters \mu and \sigma
has a support 0, 1, 2, ... and density given by
f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}
where the function _1F_1(a;c;z) is defined as
_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}
and (a)_r = \frac{\gamma(a+r)}{\gamma(a)} for a>0 and r positive integer.
This function calculates the mean and variance of this distribution.
Value
the function returns a list with the mean and variance.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Sáez-Castillo, A. J., & Conde-Sánchez, A. (2013). A hyper-Poisson regression model for overdispersed and underdispersed count data. Computational Statistics & Data Analysis, 61, 148-157.
See Also
Examples
# Example 1
# Theoretical values
mean_var_hp(mu=5.5, sigma=0.1)
# Using simulated values
y <- rHYPERPO(n=1000, mu=5.5, sigma=0.1)
mean(y)
var(y)
# Example 2
# Theoretical values
mean_var_hp2(mu=5.5, sigma=1.9)
# Using simulated values
y <- rHYPERPO2(n=1000, mu=5.5, sigma=1.9)
mean(y)
var(y)
Auxiliar function to obtain lambda from E(X) in HYPERPO2
Description
This function implements the procedure given in page 152.
Usage
obtaining_lambda(media, gamma)
Arguments
media |
the value for the mean or E(X). |
gamma |
the value for the gamma parameter. |
Value
returns the value of lambda to ensure the mean and gamma.
Draw the CDF for a discrete random variable
Description
Draw the CDF for a discrete random variable
Usage
plot_discrete_cdf(x, fx, col = "blue", lwd = 3, ...)
Arguments
x |
vector with the values of the random variable |
fx |
vector with the probabilities of |
col |
color for the line. |
lwd |
line width. |
... |
further arguments and graphical parameters. |
Value
A plot with the cumulative distribution function.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
Examples
# Example 1
# for a particular distribution
x <- 1:6
fx <- c(0.19, 0.21, 0.4, 0.12, 0.05, 0.03)
plot_discrete_cdf(x, fx, las=1, main="")
# Example 2
# for a Poisson distribution
x <- 0:10
fx <- dpois(x, lambda=3)
plot_discrete_cdf(x, fx, las=1,
main="CDF for Poisson")
The simulate_hp
Description
Auxiliar function to generate a single observation for HYPERPO.
This function is used inside random function of Hyper Poisson.
Usage
simulate_hp(sigma, mu)
simulate_hp(sigma, mu)
Arguments
sigma |
value for sigma parameter. |
mu |
value for mu parameter. |
Value
a single value for the HYPERPO distribution.
Auxiliar function for F11
Description
This function is used inside F11 function.
Usage
stopping(x, tol)
Arguments
x |
vector |
tol |
this is the tolerance of the infinite sum. |
Value
returns a logical value if the tolerance level is met.