Package 'RelDists'

Description

The Additive Weibull distribution

Usage

AddW(mu.link = "log", sigma.link = "log", nu.link = "log", tau.link = "log")

Arguments

Details

Additive Weibull distribution with parameters mu, sigma, nu and tau has density given by

$f(x) = (\mu\nu x^{\nu - 1} + \sigma\tau x^{\tau - 1}) \exp({-\mu x^{\nu} - \sigma x^{\tau} }),$

for x > 0.

Value

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

Author(s)

Amylkar Urrea Montoya, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Xie M, Lai CD (1996). “Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function.” Reliability Engineering and System Safety, 52, 83–93. doi:10.1016/0951-8320(95)00149-2.

See Also

dAddW

Examples

# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
# Will not be run this example because high number is cycles
# is needed in order to get good estimates
## Not run: 
y <- rAddW(n=100, mu=1.5, sigma=0.2, nu=3, tau=0.8)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1, family='AddW',
              control=gamlss.control(n.cyc=5000, trace=FALSE))

# Extracting the fitted values for mu, sigma, nu and tau
# using the inverse link function
exp(coef(mod, what='mu'))
exp(coef(mod, what='sigma'))
exp(coef(mod, what='nu'))
exp(coef(mod, what='tau'))

## End(Not run)

# Example 2
# Generating random values under some model
# Will not be run this example because high number is cycles
# is needed in order to get good estimates
## Not run: 
n <- 200
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(1.67 + -3 * x1)
sigma <- exp(0.69 - 2 * x2)
nu <- 3
tau <- 0.8
x <- rAddW(n=n, mu, sigma, nu, tau)

mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~1, family=AddW,
              control=gamlss.control(n.cyc=5000, trace=FALSE))

coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
exp(coef(mod, what="tau"))

## End(Not run)

Description

The Beta Generalized Exponentiated family

Usage

BGE(mu.link = "log", sigma.link = "log", nu.link = "log", tau.link = "log")

Arguments

Details

The Beta Generalized Exponentiated distribution with parameters mu, sigma, nu and tau has density given by

$f(x)= \frac{\nu \tau}{B(\mu, \sigma)} \exp(-\nu x)(1- \exp(-\nu x))^{\tau \mu - 1} (1 - (1- \exp(-\nu x))^\tau)^{\sigma -1},$

for $x > 0$, $\mu > 0$, $\sigma > 0$, $\nu > 0$ and $\tau > 0$.

Value

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

Author(s)

Johan David Marin Benjumea, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Barreto-Souza W, Santos AH, Cordeiro GM (2010). “The beta generalized exponential distribution.” Journal of Statistical Computation and Simulation, 80(2), 159–172.

See Also

dBGE

Examples

# Generating some random values with
# known mu, sigma, nu and tau
y <- rBGE(n=100, mu = 1.5, sigma =1.7, nu=1, tau=1)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1, family=BGE,
              control=gamlss.control(n.cyc=5000, trace=FALSE))

# Extracting the fitted values for mu, sigma, nu and tau
# using the inverse link function
exp(coef(mod, what='mu'))
exp(coef(mod, what='sigma'))
exp(coef(mod, what='nu'))
exp(coef(mod, what='tau'))

# Example 2
# Generating random values under some model
n <- 200
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(0.5 - x1)
sigma <- exp(0.8 - x2)
nu <- 1
tau <- 1
x <- rBGE(n=n, mu, sigma, nu, tau)

mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~1, family=BGE,
              control=gamlss.control(n.cyc=5000, trace=FALSE))

coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
exp(coef(mod, what="tau"))

Description

The Cosine Sine Exponential family

Usage

CS2e(mu.link = "log", sigma.link = "log", nu.link = "log")

Arguments

Details

The Cosine Sine Exponential distribution with parameters mu, sigma and nu has density given by

$f(x)=\frac{\pi \sigma \mu \exp(\frac{-x} {\nu})}{2 \nu [(\mu\sin(\frac{\pi}{2} \exp(\frac{-x} {\nu})) + \sigma\cos(\frac{\pi}{2} \exp(\frac{-x} {\nu}))]^2},$

for $x > 0$, $\mu > 0$, $\sigma > 0$ and $\nu > 0$.

Value

Returns a gamlss.family object which can be used to fit a CS2e distribution in the gamlss() function.

Author(s)

Johan David Marin Benjumea, [email protected]

References

Chesneau C, Bakouch HS, Hussain T (2018). “A new class of probability distributions via cosine and sine functions with applications.” Communications in Statistics-Simulation and Computation, 1–14.

See Also

dCS2e

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu 
y <- rCS2e(n=100, mu = 0.1, sigma =1, nu=0.5)

# Fitting the model
require(gamlss)

mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='CS2e',
              control=gamlss.control(n.cyc=5000, trace=FALSE))

# Extracting the fitted values for mu, sigma and nu
# using the inverse link function
exp(coef(mod, what='mu'))
exp(coef(mod, what='sigma'))
exp(coef(mod, what='nu'))

# Example 2
# Generating random values under some model
n <- 200
x1 <- runif(n, min=0.45, max=0.55)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(0.2 - x1)
sigma <- exp(0.8 - x2)
nu <- 0.5
x <- rCS2e(n=n, mu, sigma, nu)

mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1,family=CS2e,
              control=gamlss.control(n.cyc=50000, trace=FALSE))

coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))

Description

Density, distribution function, quantile function, random generation and hazard function for the Additive Weibull distribution with parameters mu, sigma, nu and tau.

Usage

dAddW(x, mu, sigma, nu, tau, log = FALSE)

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

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

rAddW(n, mu, sigma, nu, tau)

hAddW(x, mu, sigma, nu, tau)

Arguments

Details

Additive Weibull Distribution with parameters mu, sigma, nu and tau has density given by

$f(x) = (\mu\nu x^{\nu - 1} + \sigma\tau x^{\tau - 1}) \exp({-\mu x^{\nu} - \sigma x^{\tau} }),$

for x > 0.

Value

dAddW gives the density, pAddW gives the distribution function, qAddW gives the quantile function, rAddW generates random deviates and hAddW gives the hazard function.

Author(s)

Amylkar Urrea Montoya, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Xie M, Lai CD (1996). “Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function.” Reliability Engineering and System Safety, 52, 83–93. doi:10.1016/0951-8320(95)00149-2.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
curve(dAddW(x, mu=1.5, sigma=0.5, nu=3, tau=0.8), from=0.0001, to=2,
      col="red", las=1, ylab="f(x)")

## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pAddW(x, mu=1.5, sigma=0.5, nu=3, tau=0.8),
      from=0.0001, to=2, col="red", las=1, ylab="F(x)")
curve(pAddW(x, mu=1.5, sigma=0.5, nu=3, tau=0.8, lower.tail=FALSE),
      from=0.0001, to=2, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qAddW(p, mu=1.5, sigma=0.2, nu=3, tau=0.8), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pAddW(x, mu=1.5, sigma=0.2, nu=3, tau=0.8), 
      from=0, add=TRUE, col="red")

## The random function
hist(rAddW(n=10000, mu=1.5, sigma=0.2, nu=3, tau=0.8), freq=FALSE,
     xlab="x", las=1, main="")
curve(dAddW(x, mu=1.5, sigma=0.2, nu=3, tau=0.8),
      from=0.09, to=5, add=TRUE, col="red")

## The Hazard function
curve(hAddW(x, mu=1.5, sigma=0.2, nu=3, tau=0.8), from=0.001, to=1,
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the Beta Generalized Exponentiated distribution with parameters mu, sigma, nu and tau.

Usage

dBGE(x, mu, sigma, nu, tau, log = FALSE)

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

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

rBGE(n, mu, sigma, nu, tau)

hBGE(x, mu, sigma, nu, tau)

Arguments

Details

The Beta Generalized Exponentiated Distribution with parameters mu, sigma, nu and tau has density given by

$f(x)= \frac{\nu \tau}{B(\mu, \sigma)} \exp(-\nu x)(1- \exp(-\nu x))^{\tau \mu - 1} (1 - (1- \exp(-\nu x))^\tau)^{\sigma -1},$

for $x > 0$, $\mu > 0$, $\sigma > 0$, $\nu > 0$ and $\tau > 0$.

Value

dBGE gives the density, pBGE gives the distribution function, qBGE gives the quantile function, rBGE generates random deviates and hBGE gives the hazard function.

Author(s)

Johan David Marin Benjumea, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Barreto-Souza W, Santos AH, Cordeiro GM (2010). “The beta generalized exponential distribution.” Journal of Statistical Computation and Simulation, 80(2), 159–172.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function 
curve(dBGE(x, mu = 1.5, sigma =1.7, nu=1, tau=1), from = 0, to = 3, 
      col = "red", las = 1, ylab = "f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pBGE(x, mu = 1.5, sigma =1.7, nu=1, tau=1), from = 0, to = 6, 
      ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pBGE(x, mu = 1.5, sigma =1.7, nu=1, tau=1, lower.tail = FALSE), 
      from = 0, to = 6, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")

## The quantile function
p <- seq(from = 0, to = 0.99999, length.out = 100)
plot(x = qBGE(p = p, mu = 1.5, sigma =1.7, nu=1, tau=1), y = p, 
     xlab = "Quantile", las = 1, ylab = "Probability")
curve(pBGE(x, mu = (1/4), sigma =1, nu=1, tau=2), from = 0, add = TRUE, 
      col = "red")

## The random function
hist(rBGE(1000, mu = 1.5, sigma =1.7, nu=1, tau=1), freq = FALSE, xlab = "x", 
     ylim = c(0, 1), las = 1, main = "")
curve(dBGE(x, mu = 1.5, sigma =1.7, nu=1, tau=1),  from = 0, add = TRUE, 
      col = "red", ylim = c(0, 0.5))

## The Hazard function(
par(mfrow=c(1,1))
curve(hBGE(x, mu = 0.9, sigma =0.5, nu=1, tau=1), from = 0, to = 2, 
      col = "red", ylab = "Hazard function", las = 1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the Cosine Sine Exponential distribution with parameters mu, sigma and nu.

Usage

dCS2e(x, mu, sigma, nu, log = FALSE)

pCS2e(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

qCS2e(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

rCS2e(n, mu, sigma, nu)

hCS2e(x, mu, sigma, nu)

Arguments

Details

The Cosine Sine Exponential Distribution with parameters mu, sigma and nu has density given by

$f(x)=\frac{\pi \sigma \mu \exp(\frac{-x} {\nu})}{2 \nu [(\mu\sin(\frac{\pi}{2} \exp(\frac{-x} {\nu})) + \sigma\cos(\frac{\pi}{2} \exp(\frac{-x} {\nu}))]^2},$

for $x > 0$, $\mu > 0$, $\sigma > 0$ and $\nu > 0$.

Value

dCS2e gives the density, pCS2e gives the distribution function, qCS2e gives the quantile function, rCS2e generates random deviates and hCS2e gives the hazard function.

Author(s)

Juan Pablo Ramirez

References

Chesneau C, Bakouch HS, Hussain T (2018). “A new class of probability distributions via cosine and sine functions with applications.” Communications in Statistics-Simulation and Computation, 1–14.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
par(mfrow=c(1,1))
curve(dCS2e(x, mu=1, sigma=0.1, nu =0.1), from=0, to=1,
      ylim=c(0, 3), col="red", las=1, ylab="f(x)")
      
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pCS2e(x, mu=1, sigma=0.1, nu =0.1),
      from=0, to=1,  col="red", las=1, ylab="F(x)")
curve(pCS2e(x, mu=1, sigma=0.1, nu =0.1, lower.tail=FALSE),
      from=0, to=1, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qCS2e(p, mu=0.1, sigma=1, nu=0.1), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pCS2e(x, mu=0.1, sigma=1, nu=0.1), from=0, add=TRUE, col="red")

## The random function
hist(rCS2e(n=10000, mu=0.1, sigma=1, nu=0.1), freq=FALSE,
     xlab="x", las=1, main="")
curve(dCS2e(x, mu=0.1, sigma=1, nu=0.1), from=0, add=TRUE, col="red")

## The Hazard function
par(mfrow=c(1,1))
curve(hCS2e(x, mu=1, sigma=0.1, nu =0.1), from=0, to=1, ylim=c(0, 10),
      col=2, ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the Extended Exponential Geometric distribution with parameters mu and sigma.

Usage

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

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

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

rEEG(n, mu, sigma)

hEEG(x, mu, sigma)

Arguments

Details

The Extended Exponential Geometric distribution with parameters mu, and sigmahas density given by

$f(x)= \mu \sigma \exp(-\mu x)(1 - (1 - \sigma)\exp(-\mu x))^{-2},$

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

Value

dEEG gives the density, pEEG gives the distribution function, qEEG gives the quantile function, rEEG generates random deviates and hEEG gives the hazard function.

Author(s)

Johan David Marin Benjumea, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Adamidis K, Dimitrakopoulou T, Loukas S (2005). “On an extension of the exponential-geometric distribution.” Statistics & probability letters, 73(3), 259–269.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function 
par(mfrow=c(1,1))
curve(dEEG(x, mu = 1, sigma =3), from = 0, to = 10, 
      col = "red", las = 1, ylab = "f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pEEG(x, mu = 1, sigma =3), from = 0, to = 10, 
      ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pEEG(x, mu = 1, sigma =3, lower.tail = FALSE), 
      from = 0, to = 6, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")

## The quantile function
p <- seq(from = 0, to = 0.99999, length.out = 100)
plot(x = qEEG(p = p, mu = 1, sigma =0.5), y = p, 
     xlab = "Quantile", las = 1, ylab = "Probability")
curve(pEEG(x, mu = 1, sigma =0.5), from = 0, add = TRUE, 
      col = "red")

## The random function
hist(rEEG(1000, mu = 1, sigma =1), freq = FALSE, xlab = "x", 
     ylim = c(0, 0.9), las = 1, main = "")
curve(dEEG(x, mu = 1, sigma =1),  from = 0, add = TRUE, 
      col = "red", ylim = c(0, 0.8))

## The Hazard function
par(mfrow=c(1,1))
curve(hEEG(x, mu = 1, sigma =0.5), from = 0, to = 2, 
      col = "red", ylab = "Hazard function", las = 1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the four parameter Exponentiated Generalized Gamma distribution with parameters mu, sigma, nu and tau.

Usage

dEGG(x, mu, sigma, nu, tau, log = FALSE)

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

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

rEGG(n, mu, sigma, nu, tau)

hEGG(x, mu, sigma, nu, tau)

Arguments

Details

Four-Parameter Exponentiated Generalized Gamma distribution with parameters mu, sigma, nu and tau has density given by

$f(x) = \frac{\nu \sigma}{\mu \Gamma(\tau)} \left(\frac{x}{\mu}\right)^{\sigma \tau -1} \exp\left\{ - \left( \frac{x}{\mu} \right)^\sigma \right\} \left\{ \gamma_1\left( \tau, \left( \frac{x}{\mu} \right)^\sigma \right) \right\}^{\nu-1} ,$

for x > 0.

Value

dEGG gives the density, pEGG gives the distribution function, qEGG gives the quantile function, rEGG generates random deviates and hEGG gives the hazard function.

Author(s)

Amylkar Urrea Montoya, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Gauss M. C, Edwin M.M O, Giovana O. S (2011). “The exponentiated generalized gamma distribution with application to lifetime data.” Journal of Statistical Computation and Simulation, 81(7), 827–842. doi:10.1080/00949650903517874.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
curve(dEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5), from=0.000001, to=1.5, ylim=c(0, 2.5),
      col="red", las=1, ylab="f(x)")

## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5),
      from=0.000001, to=1.5, col="red", las=1, ylab="F(x)")
curve(pEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5, lower.tail=FALSE),
      from=0.000001, to=1.5, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qEGG(p, mu=0.1, sigma=0.8, nu=10, tau=1.5), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5), 
      from=0.00001, add=TRUE, col="red")

## The random function
hist(rEGG(n=100, mu=0.1, sigma=0.8, nu=10, tau=1.5), freq=FALSE,
     xlab="x", las=1, main="")
curve(dEGG(x, mu=0.1, sigma=0.8, nu=10, tau=1.5),
      from=0.0001, to=2, add=TRUE, col="red")

## The Hazard function
curve(hEGG(x,  mu=0.1, sigma=0.8, nu=10, tau=1.5), from=0.0001, to=1.5,
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the Exponentiated Modifien Weibull Extension distribution with parameters mu, sigma, nu and tau.

Usage

dEMWEx(x, mu, sigma, nu, tau, log = FALSE)

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

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

rEMWEx(n, mu, sigma, nu, tau)

hEMWEx(x, mu, sigma, nu, tau)

Arguments

Details

The Exponentiated Modifien Weibull Extension Distribution with parameters mu, sigma, nu and tau has density given by

$f(x)= \nu \sigma \tau (\frac{x}{\mu})^{\sigma-1} \exp((\frac{x}{\mu})^\sigma + \nu \mu (1- \exp((\frac{x}{\mu})^\sigma))) (1 - \exp (\nu\mu (1- \exp((\frac{x}{\mu})^\sigma))))^{\tau-1} ,$

for $x > 0$, $\nu> 0$, $\mu > 0$, $\sigma> 0$ and $\tau > 0$.

Value

dEMWEx gives the density, pEMWEx gives the distribution function, qEMWEx gives the quantile function, rEMWEx generates random deviates and hEMWEx gives the hazard function.

Author(s)

Johan David Marin Benjumea, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Sarhan AM, Apaloo J (2013). “Exponentiated modified Weibull extension distribution.” Reliability Engineering & System Safety, 112, 137–144.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function 
curve(dEMWEx(x, mu = 49.046, sigma =3.148, nu=0.00005, tau=0.1), from=0, to=100,
      col = "red", las = 1, ylab = "f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pEMWEx(x, mu = (1/4), sigma =1, nu=1, tau=2), from = 0, to = 1, 
      ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pEMWEx(x, mu = (1/4), sigma =1, nu=1, tau=2, lower.tail = FALSE), 
      from = 0, to = 1, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")

## The quantile function
p <- seq(from = 0, to = 0.99999, length.out = 100)
plot(x = qEMWEx(p = p, mu = 49.046, sigma =3.148, nu=0.00005, tau=0.1), y = p, 
     xlab = "Quantile", las = 1, ylab = "Probability")
curve(pEMWEx(x, mu = 49.046, sigma =3.148, nu=0.00005, tau=0.1), from = 0, add = TRUE, 
      col = "red")

## The random function
hist(rEMWEx(1000, mu = (1/4), sigma =1, nu=1, tau=2), freq = FALSE, xlab = "x", 
     las = 1, main = "")
curve(dEMWEx(x, mu = (1/4), sigma =1, nu=1, tau=2),  from = 0, add = TRUE, 
      col = "red", ylim = c(0, 0.5))

## The Hazard function(
par(mfrow=c(1,1))
curve(hEMWEx(x, mu = 49.046, sigma =3.148, nu=0.00005, tau=0.1), from = 0, to = 80, 
      col = "red", ylab = "Hazard function", las = 1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the Extended Odd Fr?chet-Nadarajah-Haghighi distribution with parameters mu, sigma, nu and tau.

Usage

dEOFNH(x, mu, sigma, nu, tau, log = FALSE)

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

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

rEOFNH(n, mu, sigma, nu, tau)

hEOFNH(x, mu, sigma, nu, tau)

Arguments

Details

Tthe Extended Odd Frechet-Nadarajah-Haghighi mu, sigma, nu and tau has density given by

$f(x)= \frac{\mu\sigma\nu\tau(1+\nu x)^{\sigma-1}e^{(1-(1+\nu x)^\sigma)}[1-(1-e^{(1-(1+\nu x)^\sigma)})^{\mu}]^{\tau-1}}{(1-e^{(1-(1+\nu x)^{\sigma})})^{\mu\tau+1}} e^{-[(1-e^{(1-(1+\nu x)^\sigma)})^{-\mu}-1]^{\tau}},$

for $x > 0$, $\mu > 0$, $\sigma > 0$, $\nu > 0$ and $\tau > 0$.

Value

dEOFNH gives the density, pEOFNH gives the distribution function, qEOFNH gives the quantile function, rEOFNH generates random numbers and hEOFNH gives the hazard function.

Author(s)

Helber Santiago Padilla

References

Nasiru S (2018). “Extended Odd Fréchet-G Family of Distributions.” Journal of Probability and Statistics, 2018.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

##The probability density function
par(mfrow=c(1,1))
 curve(dEOFNH(x, mu=18.5, sigma=5.1, nu=0.1, tau=0.1), from=0, to=10,
     ylim=c(0, 0.25), col="red", las=1, ylab="f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pEOFNH(x,mu=18.5, sigma=5.1, nu=0.1, tau=0.1), from = 0, to = 10, 
ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pEOFNH(x, mu=18.5, sigma=5.1, nu=0.1, tau=0.1, lower.tail = FALSE), 
from = 0, to = 10, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")

##The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qEOFNH(p, mu=18.5, sigma=5.1, nu=0.1, tau=0.1), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pEOFNH(x, mu=18.5, sigma=5.1, nu=0.1, tau=0.1), from=0, add=TRUE, col="red")

##The random function
hist(rEOFNH(n=10000, mu=18.5, sigma=5.1, nu=0.1, tau=0.1), freq=FALSE,
     xlab="x", las=1, main="")
curve(dEOFNH(x, mu=18.5, sigma=5.1, nu=0.1, tau=0.1), from=0, add=TRUE, col="red", ylim=c(0,1.25))

##The Hazard function
par(mfrow=c(1,1))
curve(hEOFNH(x, mu=18.5, sigma=5.1, nu=0.1, tau=0.1), from=0, to=10, ylim=c(0, 1),
     col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the exponentiated Weibull distribution with parameters mu, sigma and nu.

Usage

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

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

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

rEW(n, mu, sigma, nu)

hEW(x, mu, sigma, nu)

Arguments

Details

The Exponentiated Weibull Distribution with parameters mu, sigma and nu has density given by

$f(x)=\nu \mu \sigma x^{\sigma-1} \exp(-\mu x^\sigma) (1-\exp(-\mu x^\sigma))^{\nu-1},$

for $x > 0$, $\mu > 0$, $\sigma > 0$ and $\nu > 0$.

Value

dEW gives the density, pEW gives the distribution function, qEW gives the quantile function, rEW generates random deviates and hEW gives the hazard function.

See Also

EW

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
curve(dEW(x, mu=2, sigma=1.5, nu=0.5), from=0, to=2,
      ylim=c(0, 2.5), col="red", las=1, ylab="f(x)") 

## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pEW(x, mu=2, sigma=1.5, nu=0.5), 
      from=0, to=2,  col="red", las=1, ylab="F(x)")
curve(pEW(x, mu=2, sigma=1.5, nu=0.5, lower.tail=FALSE), 
      from=0, to=2, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qEW(p, mu=2, sigma=1.5, nu=0.5), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pEW(x, mu=2, sigma=1.5, nu=0.5), from=0, add=TRUE, col="red")

## The random function
hist(rEW(n=10000, mu=2, sigma=1.5, nu=0.5), freq=FALSE, 
     xlab="x", las=1, main="")
curve(dEW(x, mu=2, sigma=1.5, nu=0.5), from=0, add=TRUE, col="red") 

## The Hazard function
par(mfrow=c(1,1))
curve(hEW(x, mu=2, sigma=1.5, nu=0.5), from=0, to=2, ylim=c(0, 7), 
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the exponentiated XLindley distribution with parameters mu and sigma.

Usage

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

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

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

rEXL(n, mu, sigma)

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

Arguments

Details

The exponentiated XLindley with parameters mu and sigma has density given by

$f(x) = \frac{\sigma\mu^2(2+\mu + x)\exp(-\mu x)}{(1+\mu)^2}\left[1- \left(1+\frac{\mu x}{(1 + \mu)^2}\right) \exp(-\mu x)\right] ^ {\sigma-1}$

for $x \geq 0$, $\mu \geq 0$ and $\sigma \geq 0$.

Note: In this implementation we changed the original parameters $\delta$ for $\mu$ and $\alpha$ for $\sigma$, we did it to implement this distribution within gamlss framework.

Value

dEXL gives the density, pEXL gives the distribution function, qEXL gives the quantile function, rEXL generates random deviates and hEXL gives the hazard function.

Author(s)

Manuel Gutierrez Tangarife, [email protected]

References

Alomair, A. M., Ahmed, M., Tariq, S., Ahsan-ul-Haq, M., & Talib, J. (2024). An exponentiated XLindley distribution with properties, inference and applications. Heliyon, 10(3).

See Also

dEXL.

Examples

#Example 1
#Plotting the mass function for different parameter values
curve(dEXL(x, mu=0.5, sigma=0.5), 
      from=0.001, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dEXL(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dEXL(x, mu=1.5, sigma=0.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=0.5", 
                            "mu=1.0, sigma=0.5",
                            "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


curve(dEXL(x, mu=0.5, sigma=1), 
      from=0.001, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dEXL(x, mu=1, sigma=1),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dEXL(x, mu=1.5, sigma=1),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=1", 
                            "mu=1.0, sigma=1",
                            "mu=1.5, sigma=1"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


curve(dEXL(x, mu=0.5, sigma=1.5), 
      from=0.001, to=8,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dEXL(x, mu=1, sigma=1.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dEXL(x, mu=1.5, sigma=1.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=1.5", 
                            "mu=1.0, sigma=1.5",
                            "mu=1.5, sigma=1.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)

# Example 2
# Checking if the cumulative curves converge to 1
curve(pEXL(x, mu=0.5, sigma=0.5), 
      from=0.001, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="f(x)")
curve(pEXL(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(pEXL(x, mu=1.5, sigma=0.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=0.5, sigma=0.5", 
                               "mu=1.0, sigma=0.5",
                               "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.5)
curve(pEXL(x, mu=0.5, sigma=0.5, lower.tail=FALSE), 
      from=0.001, to=5,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="f(x)")
curve(pEXL(x, mu=1, sigma=0.5, lower.tail=FALSE),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(pEXL(x, mu=1.5, sigma=0.5, lower.tail=FALSE),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=0.5, sigma=0.5", 
                            "mu=1.0, sigma=0.5",
                            "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.5)

#example 3
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qEXL(p, mu=2.3, sigma=1.7), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pEXL(x, mu=2.3, sigma=1.7), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

#some values
p <- c(0.25, 0.5, 0.75)
quantile <- qEXL(p=p, mu=2.3, sigma=1.7) 
for(i in quantile){
  print(integrate(dEXL, lower=0, upper=i, mu=2.3, sigma=1.7))
}

#example 4
## The random function
x <- rEXL(n=10000, mu=1.5, sigma=2.5)
hist(x, freq=FALSE)
curve(dEXL(x, mu=1.5, sigma=2.5), from=0, to=20, 
      add=TRUE, col="tomato", lwd=2)

#example 5
## The Hazard function
curve(hEXL(x, mu=1.5, sigma=2), from=0.001, to=4,
      col="tomato", ylab="Hazard function", las=1)

Description

Density, distribution function, quantile function, random generation and hazard function for the Extended Weibull distribution with parameters mu, sigma and nu.

Usage

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

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

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

rExW(n, mu, sigma, nu)

hExW(x, mu, sigma, nu)

Arguments

Details

The Extended Weibull distribution with parameters mu, sigma and nu has density given by

$f(x) = \frac{\mu \sigma \nu x^{\sigma -1} exp({-\mu x^{\sigma}})} {[1 -(1-\nu) exp({-\mu x^{\sigma}})]^2},$

for x > 0.

Value

dExW gives the density, pExW gives the distribution function, qExW gives the quantile function, rExW generates random deviates and hExW gives the hazard function.

Author(s)

Amylkar Urrea Montoya, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Tieling Z, Min X (2007). “Failure Data Analysis with Extended Weibull Distribution.” Communications in Statistics - Simulation and Computation, 36, 579–592. doi:10.1080/03610910701236081.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
curve(dExW(x, mu=0.3, sigma=2, nu=0.05), from=0.0001, to=2,
      col="red", las=1, ylab="f(x)")

## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pExW(x, mu=0.3, sigma=2, nu=0.05),
      from=0.0001, to=2, col="red", las=1, ylab="F(x)")
curve(pExW(x, mu=0.3, sigma=2, nu=0.05, lower.tail=FALSE),
      from=0.0001, to=2, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qExW(p, mu=0.3, sigma=2, nu=0.05), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pExW(x, mu=0.3, sigma=2, nu=0.05), 
      from=0, add=TRUE, col="red")

## The random function
hist(rExW(n=10000, mu=0.3, sigma=2, nu=0.05), freq=FALSE,
     xlab="x", ylim=c(0, 2), las=1, main="")
curve(dExW(x, mu=0.3, sigma=2, nu=0.05),
      from=0.001, to=4, add=TRUE, col="red")

## The Hazard function
curve(hExW(x, mu=0.3, sigma=2, nu=0.05), from=0.001, to=4,
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the Flexible Weibull Extension distribution with parameters mu and sigma.

Usage

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

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

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

rFWE(n, mu, sigma)

hFWE(x, mu, sigma)

Arguments

Details

The Flexible Weibull extension with parameters mu and sigma has density given by

$f(x) = (\mu + \sigma/x^2) \exp(\mu x - \sigma/x) \exp(-\exp(\mu x-\sigma/x))$

for x>0.

Value

dFWE gives the density, pFWE gives the distribution function, qFWE gives the quantile function, rFWE generates random deviates and hFWE gives the hazard function.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function
curve(dFWE(x, mu=0.75, sigma=0.5), from=0, to=3, 
      ylim=c(0, 1.7), col="red", las=1, ylab="f(x)")

## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pFWE(x, mu=0.75, sigma=0.5), from=0, to=3, 
      col="red", las=1, ylab="F(x)")
curve(pFWE(x, mu=0.75, sigma=0.5, lower.tail=FALSE), 
      from=0, to=3, col="red", las=1, ylab="R(x)")

## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qFWE(p, mu=0.75, sigma=0.5), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pFWE(x, mu=0.75, sigma=0.5), from=0, add=TRUE, col="red")

## The random function
hist(rFWE(n=1000, mu=2, sigma=0.5), freq=FALSE, xlab="x", 
     ylim=c(0, 2), las=1, main="")
curve(dFWE(x, mu=2, sigma=0.5), from=0, to=3, add=TRUE, col="red")

## The Hazard function
par(mfrow=c(1,1))
curve(hFWE(x, mu=0.75, sigma=0.5), from=0, to=2, ylim=c(0, 2.5), 
      col="red", ylab="Hazard function", las=1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the Gamma Weibull distribution with parameters mu, sigma, nu and tau.

Usage

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

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

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

rGammaW(n, mu, sigma, nu)

hGammaW(x, mu, sigma, nu)

Arguments

Details

The Gamma Weibull Distribution with parameters mu, sigma and nu has density given by

$f(x)= \frac{\sigma \mu^{\nu}}{\Gamma(\nu)} x^{\nu \sigma - 1} \exp(-\mu x^\sigma),$

for $x > 0$, $\mu > 0$, $\sigma \geq 0$ and $\nu > 0$.

Value

dGammaW gives the density, pGammaW gives the distribution function, qGammaW gives the quantile function, rGammaW generates random deviates and hGammaW gives the hazard function.

Author(s)

Johan David Marin Benjumea, [email protected]

References

Almalki SJ, Nadarajah S (2014). “Modifications of the Weibull distribution: A review.” Reliability Engineering & System Safety, 124, 32–55. doi:10.1016/j.ress.2013.11.010.

Stacy EW, others (1962). “A generalization of the gamma distribution.” The Annals of mathematical statistics, 33(3), 1187–1192.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function 
curve(dGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, to = 6, 
      col = "red", las = 1, ylab = "f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, to = 3, 
ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pGammaW(x, mu = 0.5, sigma = 2, nu=1, lower.tail = FALSE), 
from = 0, to = 3, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")

## The quantile function
p <- seq(from = 0, to = 0.99999, length.out = 100)
plot(x = qGammaW(p = p, mu = 0.5, sigma = 2, nu=1), y = p, 
xlab = "Quantile", las = 1, ylab = "Probability")
curve(pGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, add = TRUE, 
col = "red")

## The random function
hist(rGammaW(1000, mu = 0.5, sigma = 2, nu=1), freq = FALSE, xlab = "x", 
ylim = c(0, 1), las = 1, main = "")
curve(dGammaW(x, mu = 0.5, sigma = 2, nu=1),  from = 0, add = TRUE, 
col = "red", ylim = c(0, 1))

## The Hazard function
par(mfrow=c(1,1))
curve(hGammaW(x, mu = 0.5, sigma = 2, nu=1), from = 0, to = 2, 
ylim = c(0, 1), col = "red", ylab = "Hazard function", las = 1)

par(old_par) # restore previous graphical parameters

Description

Density, distribution function, quantile function, random generation and hazard function for the generalized Gompertz distribution with parameters mu sigma and nu.

Usage

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

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

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

rGGD(n, mu, sigma, nu)

hGGD(x, mu, sigma, nu)

Arguments

Details

The Generalized Gompertz Distribution with parameters mu, sigma and nu has density given by

$f(x)= \nu \mu \exp(-\frac{\mu}{\sigma}(\exp(\sigma x - 1))) (1 - \exp(-\frac{\mu}{\sigma}(\exp(\sigma x - 1))))^{(\nu - 1)} ,$

for $x \geq 0$, $\mu > 0$, $\sigma \geq 0$ and $\nu > 0$.

Value

dGGD gives the density, pGGD gives the distribution function, qGGD gives the quantile function, rGGD generates random deviates and hGGD gives the hazard function.

Author(s)

Johan David Marin Benjumea, [email protected]

References

El-Gohary A, Alshamrani A, Al-Otaibi AN (2013). “The generalized Gompertz distribution.” Applied Mathematical Modelling, 37(1-2), 13–24.

Examples

old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters

## The probability density function 
par(mfrow = c(1, 1))
curve(dGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 4, 
      col = "red", las = 1, ylab = "f(x)")

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 4, 
      ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5, lower.tail = FALSE), 
      from = 0, to = 4, ylim = c(0, 1), col = "red", las = 1, ylab = "R(x)")

## The quantile function
p <- seq(from = 0, to = 0.99999, length.out = 100)
plot(x = qGGD(p=p, mu=1, sigma=0.3, nu=1.5), y = p, 
     xlab = "Quantile", las = 1, ylab = "Probability")
curve(pGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, add = TRUE, 
      col = "red")

## The random function
hist(rGGD(1000, mu=1, sigma=0.3, nu=1.5), freq = FALSE, xlab = "x", 
     las = 1, ylim = c(0, 0.7), main = "")
curve(dGGD(x,mu=1, sigma=0.3, nu=1.5), from = 0, to =8, add = TRUE, 
      col = "red")

## The Hazard function
par(mfrow=c(1,1))
curve(hGGD(x, mu=1, sigma=0.3, nu=1.5), from = 0, to = 3, col = "red",
      ylab = "The hazard function", las = 1)

par(old_par) # restore previous graphical parameters