Package 'ZeroOneDists'

Beta Rectangular distribution

Description

The Beta Rectangular family

Usage

BER(mu.link = "logit", sigma.link = "log", nu.link = "logit")

Arguments

mu.link	defines the mu.link, with "logit" link as the default for the mu parameter.
sigma.link	defines the sigma.link, with "log" link as the default for the sigma parameter.
nu.link	defines the nu.link, with "logit" link as the default for the nu parameter.

Details

The Beta Rectangular distribution with parameters mu, sigma and nu has density given by

$f(x| \mu, \sigma, \nu) = \nu + (1 - \nu) b(x| \mu, \sigma)$

for $0 < x < 1$, $0 < \mu < 1$, $\sigma > 0$ and $0 < \nu < 1$. The function $b(.)$ corresponds to the traditional beta distribution that can be computed by dbeta(x, shape1=mu*sigma, shape2=(1-mu)*sigma).

Value

Returns a gamlss.family object which can be used to fit a BER distribution in the gamlss() function.

Author(s)

Karina Maria Garay, [email protected]

References

Bayes, C. L., Bazán, J. L., & García, C. (2012). A new robust regression model for proportions. Bayesian Analysis, 7(4), 841-866.

See Also

dBER

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rBER(n=500, mu=0.5, sigma=10, nu=0.5)

# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=BER)

# Extracting the fitted values for mu, sigma and nu
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))

inv_logit(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
inv_logit(coef(mod1, what="nu"))

# Example 2
# Generating random values under some model

# A function to simulate a data set with Y ~ BER
gendat <- function(n) {
  x1 <- runif(n, min=0.4, max=0.6)
  x2 <- runif(n, min=0.4, max=0.6)
  x3 <- runif(n, min=0.4, max=0.6)
  mu    <- inv_logit(-0.5 + 1*x1)
  sigma <- exp(-1 + 4.8*x2)
  nu    <- inv_logit(-1 + 0.5*x3)
  y <- rBER(n=n, mu=mu, sigma=sigma, nu=nu)
  data.frame(y=y, x1=x1, x2=x2, x3=x3)
}

set.seed(1234)
datos <- gendat(n=500)

mod2 <- gamlss(y~x1, sigma.fo=~x2, nu.fo=~x3,
               family=BER, data=datos,
               control=gamlss.control(n.cyc=500, trace=TRUE))

summary(mod2)

---

Beta Rectangular distribution version 2

Description

The Beta Rectangular family

Usage

BER2(mu.link = "logit", sigma.link = "log", nu.link = "logit")

Arguments

mu.link	defines the mu.link, with "logit" link as the default for the mu parameter.
sigma.link	defines the sigma.link, with "log" link as the default for the sigma parameter.
nu.link	defines the nu.link, with "logit" link as the default for the nu parameter.

Details

The Beta Rectangular distribution with parameters mu, sigma and nu has density given by

$f(x| \mu, \sigma, \nu) = \nu + (1 - \nu) b(x| \mu, \sigma)$

for $0 < x < 1$, $0 < \mu < 1$, $\sigma > 0$ and $0 < \nu < 1$. The function $b(.)$ corresponds to the traditional beta distribution that can be computed by dbeta(x, shape1=mu*sigma, shape2=(1-mu)*sigma).

Value

Returns a gamlss.family object which can be used to fit a BER2 distribution in the gamlss() function.

References

Bayes, C. L., Bazán, J. L., & García, C. (2012). A new robust regression model for proportions. Bayesian Analysis, 7(4), 841-866.

See Also

dBER2

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rBER2(n=500, mu=0.3, sigma=7, nu=0.4)

# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=BER2,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu, sigma and nu
# using the inverse link function
inv_logit <- function(x) 1/(1 + exp(-x))

inv_logit(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))
inv_logit(coef(mod1, what="nu"))

# Example 2
# Generating random values under some model

# A function to simulate a data set with Y ~ BER2
gendat <- function(n) {
  x1 <- runif(n, min=0.4, max=0.6)
  x2 <- runif(n, min=0.4, max=0.6)
  x3 <- runif(n, min=0.4, max=0.6)
  mu    <- inv_logit(-0.5 + 1*x1)
  sigma <- exp(-1 + 4.8*x2)
  nu    <- inv_logit(-1 + 0.5*x3)
  y <- rBER2(n=n, mu=mu, sigma=sigma, nu=nu)
  data.frame(y=y, x1=x1, x2=x2, x3=x3)
}

set.seed(1234)
datos <- gendat(n=500)

mod2 <- gamlss(y~x1, sigma.fo=~x2, nu.fo=~x3,
               family=BER2, data=datos,
               control=gamlss.control(n.cyc=500, trace=TRUE))

summary(mod2)

---

Set the Right Link Function for Specified Parameter and Distribution

Description

This function is used within the distribution family specification of a GAMLSS model to define the right link for each of the parameters of the distribution. This function should not be called by the user unless he/she specify a new distribution family or wishes to change existing link functions in the parameters.

Usage

checklink(which.link = NULL, which.dist = NULL, link = NULL, link.List = NULL)

Arguments

which.link	which parameter link e.g. which.link="mu.link"
which.dist	which distribution family e.g. which.dist="Cole.Green"
link	a repetition of which.link e.g. link=substitute(mu.link)
link.List	what link function are required e.g. link.List=c("inverse", "log", "identity")

Details

This function is the same checklink fuction from gamlss.dist package.

Value

Defines the right link for each parameter.

---

Beta Rectangular distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Beta Rectangular distribution with parameters $\mu$, $\sigma$ and $\nu$.

Usage

dBER(x, mu, sigma, nu, log = FALSE)

pBER(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

qBER(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

rBER(n, mu, sigma, nu)

Arguments

x, q	vector of (non-negative integer) quantiles.
mu	vector of the mu parameter.
sigma	vector of the sigma parameter.
nu	vector of the nu parameter.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of random values to return.

Details

The Beta Rectangular distribution with parameters $\mu$, $\sigma$ and $\nu$ has a support in $(0, 1)$ and density given by

$f(x| \mu, \sigma, \nu) = \nu + (1 - \nu) b(x| \mu, \sigma)$

for $0 < x < 1$, $0 < \mu < 1$, $\sigma > 0$ and $0 < \nu < 1$. The function $b(.)$ corresponds to the traditional beta distribution that can be computed by dbeta(x, shape1=mu*sigma, shape2=(1-mu)*sigma).

Value

dBER gives the density, pBER gives the distribution function, qBER gives the quantile function, rBER generates random deviates.

Author(s)

Karina Maria Garay, [email protected]

References

Bayes, C. L., Bazán, J. L., & García, C. (2012). A new robust regression model for proportions. Bayesian Analysis, 7(4), 841-866.

See Also

BER.

Examples

# Example 1
# Plotting the density function for different parameter values
curve(dBER(x, mu=0.5, sigma=10, nu=0),
      from=0, to=1, col="green", las=1, ylab="f(x)")

curve(dBER(x, mu=0.5, sigma=10, nu=0.2),
      add=TRUE, col= "blue1")

curve(dBER(x, mu=0.5, sigma=10, nu=0.4),
      add=TRUE, col="yellow")

curve(dBER(x, mu=0.5, sigma=10, nu=0.6),
      add=TRUE, col="red")

legend("topleft", col=c("green", "blue1", "yellow", "red"),
       lty=1, bty="n",
       legend=c("mu=0.5, sigma=10, nu=0",
                "mu=0.5, sigma=10, nu=0.2",
                "mu=0.5, sigma=10, nu=0.4",
                "mu=0.5, sigma=10, nu=0.6"))


curve(dBER(x, mu=0.3, sigma=10, nu=0),
       from=0, to=1, col="green", las=1, ylab="f(x)")

curve(dBER(x, mu=0.3, sigma=10, nu=0.2),
       add=TRUE, col= "blue1")

curve(dBER(x, mu=0.3, sigma=10, nu=0.4),
       add=TRUE, col="yellow")

curve(dBER(x, mu=0.3, sigma=10, nu=0.6),
       add=TRUE, col="red")

legend("topright", col=c("green", "blue1", "yellow", "red"),
       lty=1, bty="n",
       legend=c("mu=0.5, sigma=10, nu=0",
                "mu=0.5, sigma=10, nu=0.2",
                "mu=0.5, sigma=10, nu=0.4",
                "mu=0.5, sigma=10, nu=0.6"))


# Example 2
# Checking if the cumulative curves converge to 1
curve(pBER(x, mu=0.5, sigma=10, nu=0),
       from=0, to=1, col="green", las=1, ylab="f(x)")

curve(pBER(x, mu=0.5, sigma=10, nu=0.2),
       add=TRUE, col= "blue1")

curve(pBER(x, mu=0.5, sigma=10, nu=0.4),
       add=TRUE, col="yellow")

curve(pBER(x, mu=0.5, sigma=10, nu=0.6),
       add=TRUE, col="red")

legend("topleft", col=c("green", "blue1", "yellow", "red"),
       lty=1, bty="n",
       legend=c("mu=0.5, sigma=10, nu=0",
                "mu=0.5, sigma=10, nu=0.2",
                "mu=0.5, sigma=10, nu=0.4",
                "mu=0.5, sigma=10, nu=0.6"))

# Example 3
# Checking the quantile function
mu <- 0.5
sigma <- 10
nu <- 0.4
p <- seq(from=0.01, to=0.99, length.out=100)
plot(x=qBER(p, mu=mu, sigma=sigma, nu=nu), y=p,
     xlab="Quantile", las=1, ylab="Probability")
curve(pBER(x, mu=mu, sigma=sigma, nu=nu), add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical density
x <- rBER(n= 10000, mu=0.5, sigma=10, nu=0.1)
hist(x, freq=FALSE)
curve(dBER(x, mu=0.5, sigma=10, nu=0.1),
      col="tomato", add=TRUE)

---

Beta Rectangular distribution version 2

Description

These functions define the density, distribution function, quantile function and random generation for the Beta Rectangular distribution with parameters $\mu$, $\sigma$ and $\nu$ reparameterized to ensure $E(X)=\mu$.

Usage

dBER2(x, mu, sigma, nu, log = FALSE)

pBER2(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

qBER2(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

rBER2(n, mu, sigma, nu)

Arguments

x, q	vector of (non-negative integer) quantiles.
mu	vector of the mu parameter.
sigma	vector of the sigma parameter.
nu	vector of the nu parameter.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of random values to return.

Details

The Beta Rectangular distribution with parameters $\mu$, $\sigma$ and $\nu$ has a support in $(0, 1)$ and density given by

$f(x| \mu, \sigma, \nu) = \nu + (1 - \nu) b(x| \mu, \sigma)$

for $0 < x < 1$, $0 < \mu < 1$, $\sigma > 0$ and $0 < \nu < 1$. The function $b(.)$ corresponds to the traditional beta distribution that can be computed by dbeta(x, shape1=mu*sigma, shape2=(1-mu)*sigma).

Value

dBER2 gives the density, pBER2 gives the distribution function, qBER2 gives the quantile function, rBER2 generates random deviates.

References

Bayes, C. L., Bazán, J. L., & García, C. (2012). A new robust regression model for proportions. Bayesian Analysis, 7(4), 841-866.

See Also

BER2.

Examples

# Example 1
# Plotting the density function for different parameter values
curve(dBER2(x, mu=0.5, sigma=10, nu=0),
      from=0, to=1, col="green", las=1, ylab="f(x)")

curve(dBER2(x, mu=0.5, sigma=10, nu=0.2),
      add=TRUE, col= "blue1")

curve(dBER2(x, mu=0.5, sigma=10, nu=0.4),
      add=TRUE, col="yellow")

curve(dBER2(x, mu=0.5, sigma=10, nu=0.6),
      add=TRUE, col="red")

legend("topleft", col=c("green", "blue1", "yellow", "red"),
       lty=1, bty="n",
       legend=c("mu=0.5, sigma=10, nu=0",
                "mu=0.5, sigma=10, nu=0.2",
                "mu=0.5, sigma=10, nu=0.4",
                "mu=0.5, sigma=10, nu=0.6"))


curve(dBER2(x, mu=0.3, sigma=10, nu=0),
       from=0, to=1, col="green", las=1, ylab="f(x)")

curve(dBER2(x, mu=0.3, sigma=10, nu=0.2),
       add=TRUE, col= "blue1")

curve(dBER2(x, mu=0.3, sigma=10, nu=0.4),
       add=TRUE, col="yellow")

curve(dBER2(x, mu=0.3, sigma=10, nu=0.6),
       add=TRUE, col="red")

legend("topright", col=c("green", "blue1", "yellow", "red"),
       lty=1, bty="n",
       legend=c("mu=0.3, sigma=10, nu=0",
                "mu=0.3, sigma=10, nu=0.2",
                "mu=0.3, sigma=10, nu=0.4",
                "mu=0.3, sigma=10, nu=0.6"))


# Example 2
# Checking if the cumulative curves converge to 1
curve(pBER2(x, mu=0.5, sigma=10, nu=0),
       from=0, to=1, col="green", las=1, ylab="f(x)")

curve(pBER2(x, mu=0.5, sigma=10, nu=0.2),
       add=TRUE, col= "blue1")

curve(pBER2(x, mu=0.5, sigma=10, nu=0.4),
       add=TRUE, col="yellow")

curve(pBER2(x, mu=0.5, sigma=10, nu=0.6),
       add=TRUE, col="red")

legend("topleft", col=c("green", "blue1", "yellow", "red"),
       lty=1, bty="n",
       legend=c("mu=0.5, sigma=10, nu=0",
                "mu=0.5, sigma=10, nu=0.2",
                "mu=0.5, sigma=10, nu=0.4",
                "mu=0.5, sigma=10, nu=0.6"))

# Example 3
# Checking the quantile function
mu <- 0.5
sigma <- 10
nu <- 0.4
p <- seq(from=0.01, to=0.99, length.out=100)
plot(x=qBER2(p, mu=mu, sigma=sigma, nu=nu), y=p,
     xlab="Quantile", las=1, ylab="Probability")
curve(pBER2(x, mu=mu, sigma=sigma, nu=nu), add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical density
x <- rBER2(n= 10000, mu=0.3, sigma=10, nu=0.1)
hist(x, freq=FALSE)
curve(dBER2(x, mu=0.3, sigma=10, nu=0.1),
      col="tomato", add=TRUE)

---

Unit Half Logistic-Geometry distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Unit Half Logistic-Geometry distribution with parameter $\mu$.

Usage

dUHLG(x, mu, log = FALSE)

pUHLG(q, mu, lower.tail = TRUE, log.p = FALSE)

qUHLG(p, mu, lower.tail = TRUE, log.p = FALSE)

rUHLG(n, mu)

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 $P[X <= x]$, otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of random values to return.

Details

The Unit Half Logistic-Geometry distribution with parameter $\mu$ has a support in $(0, 1)$ and density given by

$f(x| \mu) = \frac{2 \mu}{(\mu+(2-\mu)x)^2}$

for $0 < x < 1$ and $\mu > 0$.

Value

dUHLG gives the density, pUHLG gives the distribution function, qUHLG gives the quantile function, rUHLG generates random deviates.

Author(s)

Juan Diego Suarez Hernandez, [email protected]

References

Ramadan, A. T., Tolba, A. H., & El-Desouky, B. S. (2022). A unit half-logistic geometric distribution and its application in insurance. Axioms, 11(12), 676.

See Also

UHLG.

Examples

# Example 1
# Plotting the density function for different parameter values
curve(dUHLG(x, mu=0.4), from=0.01, to=0.99,
      ylim=c(0, 5), lwd=2,
      col="black", las=1, ylab="f(x)")

curve(dUHLG(x, mu=1), lwd=2,
      add=TRUE, col="red")

curve(dUHLG(x, mu=2), lwd=2,
      add=TRUE, col="green")

curve(dUHLG(x, mu=7), lwd=2,
      add=TRUE, col="blue")

legend("topright",
       col=c("black", "red", "green", "blue"),
       lty=1, bty="n", lwd=2,
       legend=c("mu=0.4",
                "mu=1",
                "mu=2",
                "mu=7"))

# Example 2
# Checking if the cumulative curves converge to 1
curve(pUHLG(x, mu=0.25), lwd=2,
      from=0.001, to=0.999, col="black", las=1, ylab="F(x)")

curve(pUHLG(x, mu=0.7), lwd=2,
      add=TRUE, col="red")

curve(pUHLG(x, mu=1.8), lwd=2,
      add=TRUE, col="green")

curve(pUHLG(x, mu=2.2), lwd=2,
      add=TRUE, col="blue")

legend("bottomright", col=c("black", "red", "green", "blue"),
       lty=1, bty="n", lwd=2,
       legend=c("mu=0.25",
                "mu=0.7",
                "mu=1.8",
                "mu=2.2"))

# Example 3
# Checking the quantile function
mu <- 2
p <- seq(from=0.01, to=0.99, length.out=100)
plot(x=qUHLG(p, mu=mu), y=p,
     xlab="Quantile", las=1, ylab="Probability")
curve(pUHLG(x, mu=mu), add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical density
x <- rUHLG(n=10000, mu=0.5)
hist(x, freq=FALSE)
curve(dUHLG(x, mu=0.5), lwd=2,
      col="tomato", add=TRUE, from=0.01, to=0.99)

---

Unit Maxwell-Boltzmann distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Unit Maxwell-Boltzmann distribution with parameter $\mu$.

Usage

dUMB(x, mu = 1, log = FALSE)

pUMB(q, mu = 1, lower.tail = TRUE, log.p = FALSE)

qUMB(p, mu, lower.tail = TRUE, log.p = FALSE)

rUMB(n = 1, 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 $P[X <= x]$, otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of random values to return.

Details

The Unit Maxwell-Boltzmann distribution with parameter $\mu$ has a support in $(0, 1)$ and density given by

$f(x| \mu) = \frac{\sqrt(2/\pi) \log^2(1/x) \exp(-\frac{\log^2(1/x)}{2\mu^2})}{\mu^3 x}$

for $0 < x < 1$ and $\mu > 0$.

Value

dUMB gives the density, pUMB gives the distribution function, qUMB gives the quantile function, rUMB generates random deviates.

Author(s)

David Villegas Ceballos, [email protected]

References

Biçer, C., Bakouch, H. S., Biçer, H. D., Alomair, G., Hussain, T., y Almohisen, A. (2024). Unit Maxwell-Boltzmann Distribution and Its Application to Concentrations Pollutant Data. Axioms, 13(4), 226.

See Also

UMB.

Examples

# Example 1
# Plotting the density function for different parameter values
curve(dUMB(x, mu=0.4), from=0, to=1,
      ylim=c(0, 12),
      col="green", las=1, ylab="f(x)")

curve(dUMB(x, mu=1),
      add=TRUE, col="blue1")

curve(dUMB(x, mu=2),
      add=TRUE, col="black")

curve(dUMB(x, mu=7),
      add=TRUE, col="red")

legend("topright",
       col=c("green", "blue1", "black", "red"),
       lty=1, bty="n",
       legend=c("mu=0.4",
                "mu=1",
                "mu=2",
                "mu=7"))

# Example 2
# Checking if the cumulative curves converge to 1
curve(pUMB(x, mu=0.25),
      from=0, to=1, col="green", las=1, ylab="F(x)")

curve(pUMB(x, mu=0.9),
      add=TRUE, col="blue1")

curve(pUMB(x, mu=1.8),
      add=TRUE, col="black")

curve(pUMB(x, mu=2.2),
      add=TRUE, col="red")

legend("bottomright", col=c("green", "blue1", "black", "red"),
       lty=1, bty="n",
       legend=c("mu=0.25",
                "mu=0.9",
                "mu=1.8",
                "mu=2.2"))

# Example 3
# Checking the quantile function
mu <- 2
p <- seq(from=0, to=1, length.out=100)
plot(x=qUMB(p, mu=mu), y=p,
     xlab="Quantile", las=1, ylab="Probability")
curve(pUMB(x, mu=mu), add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical density
x <- rUMB(n=1000, mu=0.5)
hist(x, freq=FALSE)
curve(dUMB(x, mu=0.5),
      col="tomato", add=TRUE, from=0, to=1)

---

Unit-Power Half-Normal distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Unit-Power Half-Normal distribution with parameter $\mu$ and $\sigma$.

Usage

dUPHN(x, mu, sigma, log = FALSE)

pUPHN(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)

qUPHN(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)

rUPHN(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 $P[X <= x]$, otherwise, $P[X > x]$.
p	vector of probabilities.
n	number of random values to return.

Details

The Unit-Power Half-Normal distribution with parameters $\mu$ and $\sigma$ has a support in $(0, 1)$ and density given by

$f(x| \mu, \sigma) = \frac{2\mu}{\sigma x^2} \phi(\frac{1-x}{\sigma x}) (2 \Phi(\frac{1-x}{\sigma x})-1)^{\mu-1}$

for $0 < x < 1$, $\mu > 0$ and $\sigma > 0$.

Value

dUPHN gives the density, pUPHN gives the distribution function, qUPHN gives the quantile function, rUPHN generates random deviates.

Author(s)

Juan Diego Suarez Hernandez, [email protected]

References

Santoro, K. I., Gómez, Y. M., Soto, D., & Barranco-Chamorro, I. (2024). Unit-Power Half-Normal Distribution Including Quantile Regression with Applications to Medical Data. Axioms, 13(9), 599.

See Also

dUPHN.

Examples

# Example 1
# Plotting the density function for different parameter values
curve(dUPHN(x, mu=1, sigma=1), ylim=c(0, 5),
      from=0.01, to=0.99, col="black", las=1, ylab="f(x)")

curve(dUPHN(x, mu=2, sigma=1),
      add=TRUE, col= "red")

curve(dUPHN(x, mu=3, sigma=1),
      add=TRUE, col="seagreen")

curve(dUPHN(x, mu=4, sigma=1),
      add=TRUE, col="royalblue2")

legend("topright", col=c("black", "red", "seagreen", "royalblue2"),
       lty=1, bty="n",
       legend=c("mu=1, sigma=1",
                "mu=2, sigma=1",
                "mu=3, sigma=1",
                "mu=4, sigma=1"))

# Example 2
# Checking if the cumulative curves converge to 1
curve(pUPHN(x, mu=1, sigma=1),
       from=0.01, to=0.99, col="black", las=1, ylab="F(x)")

curve(pUPHN(x, mu=2, sigma=1),
       add=TRUE, col= "red")

curve(pUPHN(x, mu=3, sigma=1),
       add=TRUE, col="seagreen")

curve(pUPHN(x, mu=4, sigma=1),
       add=TRUE, col="royalblue2")

legend("topleft", col=c("black", "red", "seagreen", "royalblue2"),
       lty=1, bty="n",
       legend=c("mu=1, sigma=1",
                "mu=2, sigma=1",
                "mu=3, sigma=1",
                "mu=4, sigma=1"))

# Example 3
# Checking the quantile function
mu <- 2
sigma <- 3
p <- seq(from=0.01, to=0.99, length.out=100)
plot(x=qUPHN(p, mu=mu, sigma=sigma), y=p,
     xlab="Quantile", las=1, ylab="Probability")
curve(pUPHN(x, mu=mu, sigma=sigma), add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical density
x <- rUPHN(n= 10000, mu=4, sigma=1)
hist(x, freq=FALSE)
curve(dUPHN(x, mu=4, sigma=1),
      col="tomato", add=TRUE, from=0.01, to=0.99)

---

Create a Link for GAMLSS families

Description

The function make.link.gamlss() is used with gamlss.family distributions in package gamlss().Given a link, it returns a link function, an inverse link function, the derivative dpar/deta where 'par' is the appropriate distribution parameter and a function for checking the domain. It differs from the usual make.link of glm() by having extra links as the logshifto1, and the own. For the use of the own link see the example bellow. show.link provides a way in which the user can identify the link functions available for each gamlss distribution. If your required link function is not available for any of the gamlss distributions you can add it in.

Usage

make.link.gamlss(link)

Arguments

link	character or numeric; one of "logit", "probit", "cloglog", "identity", "log", "sqrt", "1/mu^2", "inverse", "logshifted", "logitshifted", or number, say lambda resulting in power link $\mu^\lambda$.

Details

The own link function is added to allow the user greater flexibility. In order to used the own link function for any of the parameters of the distribution the own link should appear in the available links for this parameter. You can check this using the function show.link. If the own do not appear in the list you can create a new function for the distribution in which own is added in the list. For example the first line of the code of the binomial distribution, BI, has change from

"mstats <- checklink("mu.link", "Binomial", substitute(mu.link), c("logit", "probit", "cloglog", "log")), in version 1.0-0 of gamlss, to

"mstats <- checklink("mu.link", "Binomial", substitute(mu.link), c("logit", "probit", "cloglog", "log", "own"))

in version 1.0-1. Given that the parameter has own as an option the user needs also to define the following four new functions in order to used an own link.

i) own.linkfun

ii) own.linkinv

iii) own.mu.eta and

iv) own.valideta.

An example is given below.

Only one parameter of the distribution at a time is allowed to have its own link, (unless the same four own functions above are suitable for more that one parameter of the distribution).

Note that from gamlss version 1.9-0 the user can introduce its own link function by define an appropriate function, (see the example below).

Value

For the make.link.gamlss a list with components

linkfun: Link function function(parameter)

linkinv: Inverse link function function(eta)

mu.eta: Derivative function(eta) dparameter/deta

valideta: function(eta) TRUE if all of eta is in the domain of linkinv.

For the show.link a list with components the available links for the distribution parameters

Note

For the links involving parameters as in logshifted and logitshifted the parameters can be passed in the definition of the distribution by calling the checklink function, for example in the definition of the tau parameter in BCPE distribution the following call is made: tstats <- checklink("tau.link", "Box Cox Power Exponential", substitute(tau.link), c("logshifted", "log", "identity"), par.link = c(1))

This function is the same make.link.gamlss fuction from gamlss.dist package.

---

Unit Half Logistic-Geometry distribution

Description

The Unit Half Logistic-Geometry family

Usage

UHLG(mu.link = "log")

Arguments

mu.link

defines the mu.link, with "log" link as the default for the mu parameter.

Details

The Unit Half Logistic-Geometry distribution with parameter mu, has density given by

$f(x| \mu) = \frac{2 \mu}{(\mu+(2-\mu)x)^2}$

for $0 < x < 1$ and $\mu > 0$.

Value

Returns a gamlss.family object which can be used to fit a UHLG distribution in the gamlss() function.

Author(s)

Juan Diego Suarez Hernandez, [email protected]

References

Ramadan, A. T., Tolba, A. H., & El-Desouky, B. S. (2022). A unit half-logistic geometric distribution and its application in insurance. Axioms, 11(12), 676.

See Also

dUHLG

Examples

# Example 1
# Generating some random values with
# known mu
y <- rUHLG(n=500, mu=7)

# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=UHLG,
               control=gamlss.control(n.cyc=500, trace=FALSE))

# Extracting the fitted values for mu, sigma and nu
# 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 ~ UHLG
gendat <- function(n) {
  x1 <- runif(n, min=0.4, max=0.6)
  x2 <- runif(n, min=0.4, max=0.6)
  mu    <- exp(-0.5 + 3*x1 - 2.5*x2)
  y <- rUHLG(n=n, mu=mu)
  data.frame(y=y, x1=x1, x2=x2)
}

datos <- gendat(n=5000)

mod2 <- gamlss(y~x1+x2,
               family=UHLG, data=datos,
               control=gamlss.control(n.cyc=500, trace=TRUE))
summary(mod2)

---

Unit Maxwell-Boltzmann family

Description

The function UMB() defines the Unit Maxwell-Boltzmann distribution, a one parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

UMB(mu.link = "log")

Arguments

mu.link

defines the mu.link, with "log" link as the default for the mu.

Details

The Unit Maxwell-Boltzmann distribution with parameter $\mu$ has a support in $(0, 1)$ and density given by

$f(x| \mu) = \frac{\sqrt(2/\pi) \log^2(1/x) \exp(-\frac{\log^2(1/x)}{2\mu^2})}{\mu^3 x}$

for $0 < x < 1$ and $\mu > 0$.

Value

Returns a gamlss.family object which can be used to fit a UMB distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Biçer, C., Bakouch, H. S., Biçer, H. D., Alomair, G., Hussain, T., y Almohisen, A. (2024). Unit Maxwell-Boltzmann Distribution and Its Application to Concentrations Pollutant Data. Axioms, 13(4), 226.

See Also

dUMB

Examples

# Example 1
# Generating some random values with
# known mu
y <- rUMB(n=300, mu=0.5)

# Fitting the model
library(gamlss)
mod1 <- gamlss(y~1, family=UMB)

# 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 ~ UMB
gendat <- function(n) {
  x1 <- runif(n)
  mu <- exp(-0.5 + 1 * x1)
  y <- rUMB(n=n, mu=mu)
  data.frame(y=y, x1=x1)
}

datos <- gendat(n=300)

mod2 <- gamlss(y~x1, family=UMB, data=datos)
summary(mod2)

# Example 3
# The first dataset measured the concentration of air pollutant CO
# in Alberta, Canada from the Edmonton Central (downtown)
# Monitoring Unit (EDMU) station during 1995.
# Measurements are listed for the period 1976-1995.
# Taken from Bicer et al. (2024) page 12.

data1 <- c(0.19, 0.20, 0.20, 0.27, 0.30,
           0.37, 0.30, 0.25, 0.23, 0.23,
           0.26, 0.23, 0.19, 0.21, 0.20,
           0.22, 0.21, 0.25, 0.25, 0.19)

mod3 <- gamlss(data1 ~ 1, family=UMB)

# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod3, what="mu"))

# Extraction of the log likelihood
logLik(mod3)

# Example 4
# The second data set measured air quality monitoring of the
# annual average concentration of the pollutant benzo(a)pyrene (BaP).
# The data were obtained from the Edmonton Central (downtown)
# Monitoring Unit (EDMU) location in Alberta, Canada, in 1995.
# Taken from Bicer et al. (2024) page 12.

data2 <- c(0.22, 0.20, 0.25, 0.15, 0.38,
           0.18, 0.52, 0.27, 0.27, 0.27,
           0.13, 0.15, 0.24, 0.37, 0.20)

mod4 <- gamlss(data2 ~ 1, family=UMB)

# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod4, what="mu"))

# Extraction of the log likelihood
logLik(mod4)

# Replicating figure 5 from Bicer et al. (2024)
# Hist and estimated pdf of Data-I and Data-II
mu1 <- 0.8452875
mu2 <- 0.8593051

# Data-I
hist(data1, freq = FALSE,
     xlim = c(0, 1.0), ylim = c(0, 10),
     main = "Histogram of Data-I",
     xlab = "y", ylab = "f(y)",
     col = "burlywood1",
     border = "darkgoldenrod4")

curve(dUMB(x, mu = mu1), add = TRUE,
      col = "blue", lwd = 2)

legend("topright", legend = c("UMB"),
       col = c("blue"), lwd = 2, bty = "n")

# Data-II
hist(data2, freq = FALSE,
     xlim = c(0, 1.0), ylim = c(0, 6),
     main = "Histogram of Data-II",
     xlab = "y", ylab = "f(y)",
     col = "burlywood1",
     border = "darkgoldenrod4")

curve(dUMB(x, mu = mu2), add = TRUE,
      col = "blue", lwd = 2)

legend("topright",
       legend = c("UMB"),
       col = c("blue"),
       lwd = 2,
       bty = "n")

# Example 5
# The third dataset measured the concentration of sulphate
# in Calgary from 31 different periods during 1995.
# Taken from Bicer et al. (2024) page 13.

data3 <- c(0.048, 0.013, 0.040, 0.082, 0.073, 0.732, 0.302,
           0.728, 0.305, 0.322,  0.045, 0.261, 0.192,
           0.357, 0.022, 0.143, 0.208, 0.104, 0.330, 0.453,
           0.135, 0.114, 0.049, 0.011, 0.008, 0.037, 0.034,
           0.015, 0.028, 0.069, 0.029)

mod5 <- gamlss(data3 ~ 1, family=UMB)

# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod5, what="mu"))

# Extraction of the log likelihood
logLik(mod5)

# Example 6
# The fourth dataset measured the concentration of pollutant CO in Alberta, Canada
# from the Calgary northwest (residential) monitoring unit (CRMU) station during 1995.
# Measurements are listed  for the period 1976-95.
# Taken from Bicer et al. (2024) page 13.

data4 <- c(0.16, 0.19, 0.24, 0.25, 0.30, 0.41, 0.40,
           0.33, 0.23, 0.27, 0.30, 0.32, 0.26, 0.25,
           0.22, 0.22, 0.18, 0.18, 0.20, 0.23)

mod6 <- gamlss(data4 ~ 1, family=UMB)

# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod6, what="mu"))

# Extraction of the log likelihood
logLik(mod6)

# Replicating figure 6 from Bicer et al. (2024)
# Hist and estimated pdf of Data-III and Data-IV
mu3 <- 1.582003
mu4 <- 0.8161202

# Data-III
hist(data3, freq = FALSE,
     xlim = c(0, 1.0), ylim = c(0, 10),
     main = "Histogram of Data-III",
     xlab = "y", ylab = "f(y)",
     col = "burlywood1",
     border = "darkgoldenrod4")

curve(dUMB(x, mu = mu3), add = TRUE,
      col = "blue", lwd = 2)

legend("topright", legend = c("UMB"),
       col = c("blue"), lwd = 2, bty = "n")

# Data-IV
hist(data4, freq = FALSE,
     xlim = c(0, 1.0), ylim = c(0, 6),
     main = "Histogram of Data-IV",
     xlab = "y", ylab = "f(y)",
     col = "burlywood1",
     border = "darkgoldenrod4")

curve(dUMB(x, mu = mu4), add = TRUE,
      col = "blue", lwd = 2)

legend("topright",
       legend = c("UMB"),
       col = c("blue"),
       lwd = 2,
       bty = "n")

---

The Unit-Power Half-Normal family

Description

The function UPHN() defines the Unit-Power Half-Normal distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

UPHN(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 UPHN distribution with parameters $\mu$ and $\sigma$ has a support in $(0, 1)$ and density given by

$f(x| \mu, \sigma) = \frac{2\mu}{\sigma x^2} \phi(\frac{1-x}{\sigma x}) (2 \Phi(\frac{1-x}{\sigma x})-1)^{\mu-1}$

for $0 < x < 1$, $\mu > 0$ and $\sigma > 0$.

Value

Returns a gamlss.family object which can be used to fit a UPHN distribution in the gamlss() function.

Author(s)

Juan Diego Suarez Hernandez, [email protected]

References

Santoro, K. I., Gomez, Y. M., Soto, D., & Barranco-Chamorro, I. (2024). Unit-Power Half-Normal Distribution Including Quantile Regression with Applications to Medical Data. Axioms, 13(9), 599.

See Also

dUPHN.

Examples

# Example 1
# Generating random values with
# known mu and sigma
require(gamlss)
mu <- 1.5
sigma <- 4.0

y <- rUPHN(1000, mu, sigma)

mod1 <- gamlss(y~1, sigma.fo=~1,  family=UPHN,
               control=gamlss.control(n.cyc=5000, trace=TRUE))

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 ~ UPHN
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(0.75 - 0.69 * x1)   # Approx 1.5
  sigma <- exp(0.5 - 0.64 * x2) # Approx 1.20
  y <- rUPHN(n, mu, sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

dat <- gendat(n=2000)

mod2 <- gamlss(y~x1, sigma.fo=~x2, family=UPHN, data=dat,
               control=gamlss.control(n.cyc=5000, trace=TRUE))

summary(mod2)

---