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)

---

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).

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).

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 $\mu > 0$.

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 $\mu > 0$.

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 $\sigma > 0$.

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)

---

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

par(mfrow = c(1, 2))

# 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")

par(mfrow = c(1, 1))

# 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

par(mfrow = c(1, 2))

# 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")

par(mfrow = c(1, 1))

---

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 COMPO distribution in the gamlss() function.

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
# 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)

---