Package 'DiscreteDists'

Title: Discrete Statistical Distributions
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.
Authors: Freddy Hernandez-Barajas [aut, cre] (ORCID: <https://orcid.org/0000-0001-7459-3329>), Fernando Marmolejo-Ramos [aut] (ORCID: <https://orcid.org/0000-0003-4680-1287>), Olga Usuga-Manco [aut] (ORCID: <https://orcid.org/0000-0003-3062-1820>), Jamiu Olumoh [aut] (ORCID: <https://orcid.org/0000-0002-7371-3920>), Osho Ajayi [aut]
Maintainer: Freddy Hernandez-Barajas <[email protected]>
License: MIT + file LICENSE
Version: 1.1.2
Built: 2026-05-19 14:04:39 UTC
Source: https://github.com/fhernanb/discretedists

Help Index

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, [email protected]

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

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 μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=(1−μ+σ)(1+μ+σ)f(x | \mu, \sigma) = \frac{(1-\mu+\sigma)}{(1+\mu+\sigma)} if x=0x=0,

f(x∣μ,σ)=4μ(μ+σ−1)x−1(μ+σ+1)x+1f(x | \mu, \sigma) = 4 \mu \frac{(\mu+\sigma-1)^{x-1}}{(\mu+\sigma+1)^{x+1}} if x=1,2,...x=1, 2, ...,

with μ>0\mu > 0, σ>0\sigma > 0 and σ>∣μ−1∣\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, [email protected]

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

dBerG.

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 μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=μx(x!)σZ(μ,σ)f(x | \mu, \sigma) = \frac{\mu^x}{(x!)^{\sigma} Z(\mu, \sigma)}

with μ>0\mu > 0, σ≥0\sigma \geq 0 and

Z(μ,σ)=∑j=0∞μj(j!)σZ(\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, [email protected]

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

dCOMPO.

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)=μE(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 μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=(μ+exp⁡(σ)−12exp⁡(σ))xexp⁡(σ)(x!)exp⁡(σ)Z(μ,σ)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 μ>0\mu > 0, σ∈ℜ\sigma \in \Re and

Z(μ,σ)=∑j=0∞μj(j!)σZ(\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, [email protected]

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

dCOMPO2.

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)

Bernoulli-geometric distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Bernoulli-geometric distribution with parameters μ\mu and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The BerG distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=(1−μ+σ)(1+μ+σ)f(x | \mu, \sigma) = \frac{(1-\mu+\sigma)}{(1+\mu+\sigma)} if x=0x=0,

f(x∣μ,σ)=4μ(μ+σ−1)x−1(μ+σ+1)x+1f(x | \mu, \sigma) = 4 \mu \frac{(\mu+\sigma-1)^{x-1}}{(\mu+\sigma+1)^{x+1}} if x=1,2,...x=1, 2, ...,

with μ>0\mu > 0, σ>0\sigma > 0 and σ>∣μ−1∣\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, [email protected]

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 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 μ\mu has a support 0, 1, 2, ... and its probability mass function (pmf) is given by

f(x∣μ)=(1x+1−μx+2)μxf(x | \mu) = (\frac{1}{x+1}-\frac{\mu}{x+2})\mu^{x}

The pmf is log-convex for all values of 0<μ<10 < \mu < 1, where f(x+1;μ)f(x;μ)\frac{f(x+1;\mu)}{f(x;\mu)} is an increasing function in xx for all values of the parameter μ\mu.

Note: in this implementation we changed the original parameters λ\lambda for μ\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, [email protected]

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 COMPO distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Conway-Maxwell-Poisson distribution with parameters μ\mu and σ\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[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of random values to return.

Details

The COMPO distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=μx(x!)σZ(μ,σ)f(x | \mu, \sigma) = \frac{\mu^x}{(x!)^{\sigma} Z(\mu, \sigma)}

with μ>0\mu > 0, σ≥0\sigma \geq 0 and

Z(μ,σ)=∑j=0∞μj(j!)σZ(\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, [email protected]

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

COMPO.

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 μ\mu and σ\sigma. This parameterization was proposed by Ribeiro et al. (2020) and the main characteristic is that E(X)=μE(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[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of random values to return.

Details

The COMPO2 distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=(μ+exp⁡(σ)−12exp⁡(σ))xexp⁡(σ)(x!)exp⁡(σ)Z(μ,σ)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 μ>0\mu > 0, σ∈ℜ\sigma \in \Re and

Z(μ,σ)=∑j=0∞μj(j!)σZ(\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, [email protected]

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

COMPO2.

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 μ\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[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of random values to return

Details

The Discrete Burr-Hatke distribution with parameters μ\mu has a support 0, 1, 2, ... and density given by

f(x∣μ)=(1x+1−μx+2)μxf(x | \mu) = (\frac{1}{x+1}-\frac{\mu}{x+2})\mu^{x}

The pmf is log-convex for all values of 0<μ<10 < \mu < 1, where f(x+1;μ)f(x;μ)\frac{f(x+1;\mu)}{f(x;\mu)} is an increasing function in xx for all values of the parameter μ\mu.

Note: in this implementation we changed the original parameters λ\lambda for μ\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, [email protected]

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 μ\mu and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The DGEII distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=(1−μx+1)σ−(1−μx)σf(x | \mu, \sigma) = (1-\mu^{x+1})^{\sigma}-(1-\mu^x)^{\sigma}

with 0<μ<10 < \mu < 1 and σ>0\sigma > 0. If σ=1\sigma=1, the DGEII distribution reduces to the geometric distribution with success probability 1−μ1-\mu.

Note: in this implementation we changed the original parameters pp to μ\mu and α\alpha to σ\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, [email protected]

References

Nekoukhou, V., Alamatsaz, M. H., & Bidram, H. (2013). Discrete generalized exponential distribution of a second type. Statistics, 47(4), 876-887.

See Also

DGEII.

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 μ\mu and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The discrete Inverted Kumaraswamy distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ... and density given by

f(x∣μ,σ)=(1−(2+x)−μ)σ−(1−(1+x)−μ)σf(x | \mu, \sigma) = (1-(2+x)^{-\mu})^{\sigma}-(1-(1+x)^{-\mu})^{\sigma}

with μ>0\mu > 0 and σ>0\sigma > 0.

Note: in this implementation we changed the original parameters α\alpha and β\beta for μ\mu and σ\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, [email protected]

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

DIKUM.

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 μ\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[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of random values to return.

Details

The Discrete Lindley distribution with parameters μ\mu has a support 0, 1, 2, ... and density given by

f(x∣μ)=e−μx1+μ[μ(1−2e−μ)+(1−e−μ)(1+μx)]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 θ\theta for μ\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, [email protected]

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 μ\mu and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The DMOLBE distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=σ((1+x/μ)exp⁡(−x/μ)−(1+(x+1)/μ)exp⁡(−(x+1)/μ))(1−(1−σ)(1+x/μ)exp⁡(−x/μ))((1−(1−σ)(1+(x+1)/μ)exp⁡(−(x+1)/μ))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 μ>0\mu > 0 and σ>0\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, [email protected]

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

DMOLBE.

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 μ\mu and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The discrete Perks distribution with parameters μ>0\mu > 0 and σ>0\sigma > 0 has a support 0, 1, 2, ... and density given by

f(x∣μ,σ)=μ(1+μ)(eσ−1)eσx(1+μeσx)(1+μeσ(x+1))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 λ\lambda for μ\mu and β\beta for σ\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, [email protected]

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

DPERKS.

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 μ\mu and σ\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[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of random values to return.

Details

The DsPA distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ...

Note:in this implementation we changed the original parameters β\beta and λ\lambda for μ\mu and σ\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, [email protected]

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 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 μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=(1−μx+1)σ−(1−μx)σf(x | \mu, \sigma) = (1-\mu^{x+1})^{\sigma}-(1-\mu^x)^{\sigma}

with 0<μ<10 < \mu < 1 and σ>0\sigma > 0. If σ=1\sigma=1, the DGEII distribution reduces to the geometric distribution with success probability 1−μ1-\mu.

Note: in this implementation we changed the original parameters pp to μ\mu and α\alpha to σ\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, [email protected]

References

Nekoukhou, V., Alamatsaz, M. H., & Bidram, H. (2013). Discrete generalized exponential distribution of a second type. Statistics, 47(4), 876-887.

See Also

dDGEII.

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 GGEO distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Generalized Geometric distribution with parameters μ\mu and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The GGEO distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=σμx(1−μ)(1−(1−σ)μx+1)(1−(1−σ)μx)f(x | \mu, \sigma) = \frac{\sigma \mu^x (1-\mu)}{(1-(1-\sigma) \mu^{x+1})(1-(1-\sigma) \mu^{x})}

with 0<μ<10 < \mu < 1 and σ>0\sigma > 0. If σ=1\sigma=1, the GGEO distribution reduces to the geometric distribution with success probability 1−μ1-\mu.

Note: in this implementation we changed the original parameters θ\theta for μ\mu and α\alpha for σ\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, [email protected]

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 μ\mu and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The hyper-Poisson distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ... and density given by

f(x∣μ,σ)=μx1F1(1;μ;σ)Γ(σ)Γ(x+σ)f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}

where the function 1F1(a;c;z)_1F_1(a;c;z) is defined as

1F1(a;c;z)=∑r=0∞(a)r(c)rzrr!_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}

and (a)r=γ(a+r)γ(a)(a)_r = \frac{\gamma(a+r)}{\gamma(a)} for a>0a>0 and rr positive integer.

Note: in this implementation we changed the original parameters λ\lambda and γ\gamma for μ\mu and σ\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, [email protected]

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

HYPERPO.

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 μ\mu (as mean) and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return
p

vector of probabilities.

Details

The hyper-Poisson distribution with parameters μ\mu and σ\sigma has a support 0, 1, 2, ...

Note: in this implementation the parameter μ\mu is the mean of the distribution and σ\sigma corresponds to the dispersion parameter. If you fit a model with this parameterization, the time will increase because an internal procedure to convert μ\mu to λ\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, [email protected]

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

HYPERPO2, HYPERPO.

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)")

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 μ\mu and σ\sigma has a support 0, 1, 2, ... and density given by

f(x∣μ,σ)=(1−(2+x)−μ)σ−(1−(1+x)−μ)σf(x | \mu, \sigma) = (1-(2+x)^{-\mu})^{\sigma}-(1-(1+x)^{-\mu})^{\sigma}

with μ>0\mu > 0 and σ>0\sigma > 0.

Note: in this implementation we changed the original parameters α\alpha and β\beta for μ\mu and σ\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, [email protected]

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

dDIKUM.

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 μ>0\mu > 0 has a support 0, 1, 2, ... and density given by

f(x∣μ)=e−μx1+μ(μ(1−2e−μ)+(1−e−μ)(1+μx))f(x | \mu) = \frac{e^{-\mu x}}{1 + \mu} (\mu(1 - 2e^{-\mu}) + (1- e^{-\mu})(1+\mu x))

The parameter μ\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 θ\theta for μ\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, [email protected]

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 μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=σ((1+x/μ)exp⁡(−x/μ)−(1+(x+1)/μ)exp⁡(−(x+1)/μ))(1−(1−σ)(1+x/μ)exp⁡(−x/μ))((1−(1−σ)(1+(x+1)/μ)exp⁡(−(x+1)/μ))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 μ>0\mu > 0 and σ>0\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, [email protected]

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

dDMOLBE.

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

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 μ\mu and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The Poisson-generalised Lindley distribution with parameters μ\mu and σ\sigma has support x=0,1,2,…x = 0, 1, 2, \ldots and probability mass function given by

f(x∣μ,σ)=μ2+μσ(μ+1)1−σΓ(x+σ)Γ(σ)Γ(x+1)(μ+1)x+2f(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 μ>0\mu > 0 and σ>0\sigma > 0.

This distribution is useful for modeling over-dispersed count data.

Note: in this implementation we changed the original parameters θ\theta and α\alpha for μ\mu and σ\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, [email protected]

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 μ\mu (as mean) and σ\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 P[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
n

number of random values to return.
p

vector of probabilities.

Details

The Poisson-generalised Lindley distribution with parameters μ\mu and σ\sigma has support x=0,1,2,…x = 0, 1, 2, \ldots and probability mass function given by

Note: in this implementation the parameter μ\mu is the mean of the distribution and σ\sigma (equivalent to α\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, [email protected]

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, NPGL2.

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 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 μ>0\mu > 0 and σ>0\sigma > 0 has a support 0, 1, 2, ... and density given by

f(x∣μ,σ)=μ(1+μ)(eσ−1)eσx(1+μeσx)(1+μeσ(x+1))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 λ\lambda for μ\mu and β\beta for σ\sigma, we did it to implement this distribution within gamlss framework.

Author(s)

Veronica Seguro Varela, [email protected]

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

dDPERKS.

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"))

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 μ\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[X<=x]P[X <= x], otherwise, P[X > x].
p

vector of probabilities.
n

number of random values to return

Details

The Discrete Poisson XLindley distribution with parameters μ\mu has a support 0, 1, 2, ... and mass function given by

f(x∣μ)=μ2(x+μ2+3(1+μ))(1+μ)4+xf(x | \mu) = \frac{\mu^2(x+\mu^2+3(1+\mu))}{(1+\mu)^{4+x}}; with μ>0\mu>0.

Note: in this implementation we changed the original parameters α\alpha for μ\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, [email protected]

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

POISXL.

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 μ\mu and σ\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[X<=x]P[X <= x], otherwise, P[X>x]P[X > x].
p

vector of probabilities.
n

number of random values to return.

Details

The Poisson–transmuted record type exponential distribution with parameters μ\mu and σ\sigma has support x=0,1,2,…x = 0,1,2,\dots and probability mass function given by

f(x∣μ,σ)=μ(1+μ)x+1(σμ(1+x)1+μ−(σ−1))f(x | \mu, \sigma) = \frac{\mu}{(1+\mu)^{x+1}} \left(\frac{\sigma \mu (1+x)}{1+\mu} - (\sigma-1) \right)

with μ>0\mu > 0 and 0<σ<10 < \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, [email protected]

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

PTRTE.

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)")

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 μ\mu and σ\sigma has a support 0, 1, 2, ... and density given by

f(x∣μ,σ)=(σx)μ(1−xμlog⁡(λ))−(σ(x+1))μ(1−(x+1)μlog⁡(λ))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 β\beta and λ\lambda for μ\mu and σ\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, [email protected]

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

dDsPA.

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"))

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_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)

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)

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 μ\mu and σ\sigma has a support 0, 1, 2, ... and mass function given by

f(x∣μ,σ)=σμx(1−μ)(1−(1−σ)μx+1)(1−(1−σ)μx)f(x | \mu, \sigma) = \frac{\sigma \mu^x (1-\mu)}{(1-(1-\sigma) \mu^{x+1})(1-(1-\sigma) \mu^{x})}

with 0<μ<10 < \mu < 1 and σ>0\sigma > 0. If σ=1\sigma=1, the GGEO distribution reduces to the geometric distribution with success probability 1−μ1-\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, [email protected]

References

Gómez-Déniz, E. (2010). Another generalization of the geometric distribution. Test, 19, 399-415.

See Also

dGGEO.

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"))

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

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 μ\mu and σ\sigma has a support 0, 1, 2, ... and density given by

f(x∣μ,σ)=μx1F1(1;μ;σ)Γ(σ)Γ(x+σ)f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}

where the function 1F1(a;c;z)_1F_1(a;c;z) is defined as

1F1(a;c;z)=∑r=0∞(a)r(c)rzrr!_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}

and (a)r=γ(a+r)γ(a)(a)_r = \frac{\gamma(a+r)}{\gamma(a)} for a>0a>0 and rr positive integer.

Note: in this implementation we changed the original parameters λ\lambda and γ\gamma for μ\mu and σ\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, [email protected]

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

dHYPERPO.

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 μ\mu and σ\sigma has a support 0, 1, 2, ...

Note: in this implementation the parameter μ\mu is the mean of the distribution and σ\sigma corresponds to the dispersion parameter. If you fit a model with this parameterization, the time will increase because an internal procedure to convert μ\mu to λ\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, [email protected]

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

dHYPERPO2, HYPERPO.

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)

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)

Mean and variance for hyper-Poisson distribution

Description

This function calculates the mean and variance for the hyper-Poisson distribution with parameters μ\mu and σ\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 μ\mu and σ\sigma has a support 0, 1, 2, ... and density given by

f(x∣μ,σ)=μx1F1(1;μ;σ)Γ(σ)Γ(x+σ)f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}

where the function 1F1(a;c;z)_1F_1(a;c;z) is defined as

1F1(a;c;z)=∑r=0∞(a)r(c)rzrr!_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}

and (a)r=γ(a+r)γ(a)(a)_r = \frac{\gamma(a+r)}{\gamma(a)} for a>0a>0 and rr 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, [email protected]

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

HYPERPO.

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)

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 μ\mu and σ\sigma has support x=0,1,2,…x = 0, 1, 2, \ldots and probability mass function given by

f(x∣μ,σ)=μ2+μσ(μ+1)1−σΓ(x+σ)Γ(σ)Γ(x+1)(μ+1)x+2f(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 μ>0\mu > 0 and σ>0\sigma > 0.

This distribution is useful for modeling over-dispersed count data.

Note: in this implementation we changed the original parameters θ\theta and α\alpha for μ\mu and σ\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, [email protected]

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

dNPGL.

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:

  • μ>0\mu > 0: the mean of the distribution, i.e. E[X]E[X].

  • σ>0\sigma > 0

The reparametrization links the mean and σ\sigma to the original parameter μ\mu via the mean equation E[X]=(σ+μ)/(μ(1+μ))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, [email protected]

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

dNPGL2, NPGL.

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.

library(COUNT)
data(azpro)
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.

library(AER)
data("NMES1988")

# 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")

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 XX.
fx

vector with the probabilities of XX.
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, [email protected]

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 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 μ\mu has a support 0, 1, 2, ... and mass function given by

f(x∣μ)=μ2(x+μ2+3(1+μ))(1+μ)4+xf(x | \mu) = \frac{\mu^2(x+\mu^2+3(1+\mu))}{(1+\mu)^{4+x}}; with μ>0\mu>0.

Note: in this implementation we changed the original parameters α\alpha for μ\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, [email protected]

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

dPOISXL.

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 μ\mu and σ\sigma has support x=0,1,2,…x = 0,1,2,\dots and probability mass function given by

f(x∣μ,σ)=μ(1+μ)x+1(σμ(1+x)1+μ−(σ−1))f(x | \mu, \sigma) = \frac{\mu}{(1+\mu)^{x+1}} \left(\frac{\sigma \mu (1+x)}{1+\mu} - (\sigma-1) \right)

Parameter restrictions: μ>0\mu > 0 and 0<σ<10 < \sigma < 1.

Note: we renamed the original parameters θ\theta and pp to μ\mu and σ\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, [email protected]

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

dPTRTE.

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"))