| Title: | Estimation for some Reliability Distributions |
| Version: | 1.0.0 |
| Description: | Parameters estimation and linear regression models for Reliability distributions families reviewed by Almalki & Nadarajah (2014) <doi:10.1016/j.ress.2013.11.010> using Generalized Additive Models for Location, Scale and Shape, aka GAMLSS by Rigby & Stasinopoulos (2005) <doi:10.1111/j.1467-9876.2005.00510.x>. |
| Depends: | R (≥ 3.5.0), survival, EstimationTools (≥ 4.0.0) |
| License: | GPL-3 |
| URL: | https://ousuga.github.io/RelDists/ |
| BugReports: | https://github.com/ousuga/RelDists/issues |
| Encoding: | UTF-8 |
| RdMacros: | Rdpack |
| Imports: | gamlss, gamlss.dist, Rdpack, zipfR, BBmisc, lamW, VGAM |
| LazyData: | true |
| Suggests: | knitr, rmarkdown, viridis, autoimage, gamlss.cens, V8 |
| RoxygenNote: | 7.2.3 |
| VignetteBuilder: | knitr |
| NeedsCompilation: | no |
| Packaged: | 2022-12-21 20:36:34 UTC; jaime |
| Author: | Freddy Hernandez 
[aut],
Olga Usuga [aut],
Carmen Patino [aut],
Jaime Mosquera
[aut, cre],
Amylkar Urrea [aut] |
| Maintainer: | Jaime Mosquera <jmosquerag@unal.edu.co> |
| Repository: | CRAN |
| Date/Publication: | 2022-12-22 20:00:10 UTC |
The Additive Weibull family
Description
The Additive Weibull distribution
Usage
AddW(mu.link = "log", sigma.link = "log", nu.link = "log", tau.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
tau.link |
defines the tau.link, with "log" link as the default for the tau parameter. |
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, amylkar.urrea@udea.edu.co
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
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)
The Beta Generalized Exponentiated family
Description
The Beta Generalized Exponentiated family
Usage
BGE(mu.link = "log", sigma.link = "log", nu.link = "log", tau.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
tau.link |
defines the tau.link, with "log" link as the default for the tau parameter. |
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, johand.marin@udea.edu.co
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
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"))
The Cosine Sine Exponential family
Description
The Cosine Sine Exponential family
Usage
CS2e(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
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, johand.marin@udea.edu.co
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
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"))
The Extended Exponential Geometric family
Description
The Extended Exponential Geometric family
Usage
EEG(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 Extended Exponential Geometric distribution with parameters mu
and sigma has 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
Returns a gamlss.family object which can be used to fit a EEG distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
See Also
Examples
# Generating some random values with
# known mu, sigma, nu and tau
y <- rEEG(n=100, mu = 1, sigma =1.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, family=EEG,
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'))
# Example 2
# Generating random values under some model
n <- 200
x1 <- runif(n, min=0.1, max=0.2)
x2 <- runif(n, min=0.1, max=0.15)
mu <- exp(0.75 - x1)
sigma <- exp(0.5 - x2)
x <- rEEG(n=n, mu, sigma)
mod <- gamlss(x~x1, sigma.fo=~x2, family=EEG,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
The four parameter Exponentiated Generalized Gamma family
Description
The four parameter Exponentiated Generalized Gamma distribution
Usage
EGG(mu.link = "log", sigma.link = "log", nu.link = "log", tau.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
tau.link |
defines the tau.link, with "log" link as the default for the tau parameter. |
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
Returns a gamlss.family object which can be used to fit a EGG distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rEGG(n=500, mu=0.1, sigma=0.8, nu=10, tau=1.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1,
family='EGG',
control=gamlss.control(n.cyc=500, 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=0.2, max=0.8)
x2 <- runif(n, min=0.2, max=0.8)
mu <- exp(-0.8 + -3 * x1)
sigma <- exp(0.77 - 2 * x2)
nu <- 10
tau <- 1.5
y <- rEGG(n=n, mu, sigma, nu, tau)
mod <- gamlss(y~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~1, family=EGG,
control=gamlss.control(n.cyc=500, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
exp(coef(mod, what="tau"))
The Exponentiated Modifien Weibull Extension family
Description
The Exponentiated Modifien Weibull Extension family
Usage
EMWEx(mu.link = "log", sigma.link = "log", nu.link = "log", tau.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
tau.link |
defines the tau.link, with "log" link as the default for the tau parameter. |
Details
The Beta-Weibull 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
Returns a gamlss.family object which can be used to fit a EMWEx distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rEMWEx(n=100, mu = 1, sigma =1.21, nu=1, tau=2)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1, family=EMWEx,
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(0.75 - x1)
sigma <- exp(0.5 - x2)
nu <- 1
tau <- 2
x <- rEMWEx(n=n, mu, sigma, nu, tau)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~1, family=EMWEx,
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"))
The Extended Odd Frechet-Nadarjad-Hanhighi family
Description
The Extended Odd Frechet-Nadarjad-Hanhighi family
Usage
EOFNH(mu.link = "log", sigma.link = "log", nu.link = "log", tau.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
tau.link |
defines the tau.link, with "log" link as the default for the tau parameter. |
Details
The Extended Odd Frechet-Nadarjad-Hanhighi distribution with parameters 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
Returns a gamlss.family object which can be used to fit a EOFNH distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
References
Nasiru S (2018). “Extended Odd Fréchet-G Family of Distributions.” Journal of Probability and Statistics, 2018.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rEOFNH(n=100, mu=1, sigma=2.1, nu=0.8, tau=1)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1, family=EOFNH,
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 <- 0.5
x <- rEOFNH(n=n, mu, sigma, nu, tau)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~1, family=EOFNH,
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"))
The Exponentiated Weibull family
Description
The Exponentiated Weibull distribution
Usage
EW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
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.
Value
Returns a gamlss.family object which can be used to fit a EW distribution in the gamlss() function.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
# Will not be run this example because high number is cycles
# is needed in order to get good estimates
## Not run:
y <- rEW(n=100, mu=2, sigma=1.5, nu=0.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='EW',
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'))
## 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 <- rpois(n, lambda=2)
x2 <- runif(n)
mu <- exp(2 + -3 * x1)
sigma <- exp(3 - 2 * x2)
nu <- 2
x <- rEW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=EW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
## End(Not run)
The Extended Weibull family
Description
The Extended Weibull family
Usage
ExW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
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
Returns a gamlss.family object which can be used to fit a ExW distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rExW(n=200, mu=0.3, sigma=2, nu=0.05)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='ExW',
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 <- 500
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-2 + 3 * x1)
sigma <- exp(1.3 - 2 * x2)
nu <- 0.05
x <- rExW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=ExW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Flexible Weibull Extension family
Description
The function FWE() defines the Flexible Weibull distribution, a two parameter
distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
Usage
FWE(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 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
Returns a gamlss.family object which can be used to fit a FWE distribution in the gamlss() function.
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rFWE(n=100, mu=0.75, sigma=1.3)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, family='FWE',
control=gamlss.control(n.cyc=5000, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod, what='mu'))
exp(coef(mod, what='sigma'))
# Example 2
# Generating random values under some model
n <- 200
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(1.21 - 3 * x1)
sigma <- exp(1.26 - 2 * x2)
x <- rFWE(n=n, mu, sigma)
mod <- gamlss(x~x1, sigma.fo=~x2, family=FWE,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
The Generalized Gompertz family
Description
The Generalized Gompertz family
Usage
GGD(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
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
Returns a gamlss.family object which can be used to fit a GGD distribution in the gamlss() function.
.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
References
El-Gohary A, Alshamrani A, Al-Otaibi AN (2013). “The generalized Gompertz distribution.” Applied Mathematical Modelling, 37(1-2), 13–24.
See Also
Examples
#Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rGGD(n=1000, mu=1, sigma=0.3, nu=1.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='GGD',
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=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(0.5 - x1)
sigma <- exp(-1 - x2)
nu <- 1.5
x <- rGGD(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=GGD,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Generalized Inverse Weibull family
Description
The Generalized Inverse Weibull family
Usage
GIW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Generalized Inverse Weibull distribution with parameters mu,
sigma and nu has density given by
f(x) = \nu \sigma \mu^{\sigma} x^{-(\sigma + 1)} exp \{-\nu (\frac{\mu}{x})^{\sigma}\},
for x > 0.
Value
Returns a gamlss.family object which can be used to fit a GIW distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
References
Felipe R SdG, Edwin M MO, Gauss M C (2009). “The generalized inverse Weibull distribution.” Statistical papers, 52(3), 591–619. doi:10.1007/s00362-009-0271-3.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rGIW(n=200, mu=3, sigma=5, nu=0.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='GIW',
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 <- 500
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-1.02 + 3 * x1)
sigma <- exp(1.69 - 2 * x2)
nu <- 0.5
x <- rGIW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=GIW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Generalized Modified Weibull family
Description
The Generalized modified Weibull distribution
Usage
GMW(mu.link = "log", sigma.link = "log", nu.link = "sqrt", tau.link = "sqrt")
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. |
nu.link |
defines the nu.link, with "sqrt" link as the default for the nu parameter. |
tau.link |
defines the tau.link, with "sqrt" link as the default for the tau parameter. |
Details
The Generalized modified Weibull distribution with parameters mu,
sigma, nu and tau has density given by
f(x)= \mu \sigma x^{\nu - 1}(\nu + \tau x) \exp(\tau x - \mu x^{\nu} e^{\tau x})
[1 - \exp(- \mu x^{\nu} e^{\tau x})]^{\sigma-1},
for x > 0.
Value
Returns a gamlss.family object which can be used to fit a GMW distribution in the gamlss() function.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rGMW(n=100, mu=2, sigma=0.5, nu=2, tau=1.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~ 1, family='GMW',
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'))
(coef(mod, what='nu'))^2
(coef(mod, what='tau'))^2
# Example 2
# Generating random values under some model
## Not run:
n <- 1000
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(2 + -3 * x1)
sigma <- exp(3 - 2 * x2)
nu <- 2
tau <- 1.5
x <- rGMW(n=n, mu, sigma, nu, tau)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~ 1, family="GMW",
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
coef(mod, what="nu")^2
coef(mod, what="tau")^2
## End(Not run)
The Gamma Weibull family
Description
The Gamma Weibull family
Usage
GammaW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
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
Returns a gamlss.family object which can be used to fit a GammaW distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rGammaW(n=100, mu = 0.5, sigma = 2, nu=1)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='GammaW',
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)
x2 <- runif(n)
mu <- exp(-1.6 * x1)
sigma <- exp(1.1 - 1 * x2)
nu <- 1
x <- rGammaW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, mu.fo=~x1, sigma.fo=~x2, nu.fo=~1, family=GammaW,
control=gamlss.control(n.cyc=50000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
coef(mod, what='nu')
The Inverse Weibull family
Description
The Inverse Weibull distribution
Usage
IW(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 Inverse Weibull distribution with parameters mu,
sigma has density given by
f(x) = \mu \sigma x^{-\sigma-1} \exp(\mu x^{-\sigma})
for x > 0, \mu > 0 and \sigma > 0
Value
Returns a gamlss.family object which can be used to fit a IW distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
Drapella A (1993). “The complementary Weibull distribution: unknown or just forgotten?” Quality and Reliability Engineering International, 9(4), 383–385.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rIW(n=100, mu=5, sigma=2.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, mu.fo=~1, sigma.fo=~1, family='IW',
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'))
# Example 2
# Generating random values under some model
n <- 200
x1 <- rpois(n, lambda=2)
x2 <- runif(n)
mu <- exp(2 + -1 * x1)
sigma <- exp(2 - 2 * x2)
x <- rIW(n=n, mu, sigma)
mod <- gamlss(x~x1, mu.fo=~1, sigma.fo=~x2, family=IW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
The Kumaraswamy Inverse Weibull family
Description
The Kumaraswamy Inverse Weibull family
Usage
KumIW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Kumaraswamy Inverse Weibull Distribution with parameters mu,
sigma and nu has density given by
f(x)= \mu \sigma \nu x^{-\mu - 1} \exp{- \sigma x^{-\mu}} (1 - \exp{- \sigma x^{-\mu}})^{\nu - 1},
for x > 0, \mu > 0, \sigma > 0 and \nu > 0.
Value
Returns a gamlss.family object which can be used to fit a KumIW distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
Shahbaz MQ, Shahbaz S, Butt NS (2012). “The Kumaraswamy-Inverse Weibull Distribution.” Shahbaz, MQ, Shahbaz, S., & Butt, NS (2012). The Kumaraswamy–Inverse Weibull Distribution. Pakistan journal of statistics and operation research, 8(3), 479–489.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rKumIW(n=1000, mu = 1.5, sigma= 1.5, nu = 5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='KumIW',
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=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(1 - x1)
sigma <- exp(1 - x2)
nu <- 5
x <- rKumIW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=KumIW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Lindley family
Description
The function LIN() defines the Lindley distribution with only one parameter
for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
Usage
LIN(mu.link = "log")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
Details
The Lindley with parameter mu has density given by
f(x) = \frac{\mu^2}{\mu+1} (1+x) \exp(-\mu x),
for x > 0 and \mu > 0.
Value
Returns a gamlss.family object which can be used to fit a LIN distribution in the gamlss() function.
Author(s)
Freddy Hernandez fhernanb@unal.edu.co
References
Lindley DV (1958). “Fiducial distributions and Bayes' theorem.” Journal of the Royal Statistical Society. Series B (Methodological), 102–107.
Lindley DV (1965). Introduction to probability and statistics: from a Bayesian viewpoint. 2. Inference. CUP Archive.
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rLIN(n=200, mu=2)
# Fitting the model
require(gamlss)
mod <- gamlss(y ~ 1, family="LIN")
# Extracting the fitted values for mu
# using the inverse link function
exp(coef(mod, what='mu'))
# Example 2
# Generating random values under some model
n <- 100
x1 <- runif(n=n)
x2 <- runif(n=n)
eta <- 1 + 3 * x1 - 2 * x2
mu <- exp(eta)
y <- rLIN(n=n, mu=mu)
mod <- gamlss(y ~ x1 + x2, family=LIN)
coef(mod, what='mu')
The Log-Weibull family
Description
The Log-Weibull distribution
Usage
LW(mu.link = "identity", 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 Log-Weibull Distribution with parameters mu
and sigma has density given by
f(y)=(1/\sigma) e^{((y - \mu)/\sigma)} exp\{-e^{((y - \mu)/\sigma)}\},
for - infty < y < infty.
Value
Returns a gamlss.family object which can be used to fit a LW distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
E.J G (1958). Statistics of extremes. Columbia University Press. ISBN 10:0231021909.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rLW(n=100, mu=0, sigma=1.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, family= 'LW',
control=gamlss.control(n.cyc=5000, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
coef(mod, 'mu')
exp(coef(mod, 'sigma'))
# 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 <- 1.5 - 3 * x1
sigma <- exp(1.4 - 2 * x2)
x <- rLW(n=n, mu, sigma)
mod <- gamlss(x~x1, sigma.fo=~x2, family=LW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
The Marshall-Olkin Extended Inverse Weibull family
Description
The Marshall-Olkin Extended Inverse Weibull family
Usage
MOEIW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Marshall-Olkin Extended Inverse 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}}\}}{\{\nu -(\nu-1) exp\{{-\mu x ^{-\sigma}}\} \}^{2}},
for x > 0.
Value
Returns a gamlss.family object which can be used to fit a MOEIW distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
References
Hassan M O, A.H E, A.M.K T, Abdulkareem M Bc (2017). “Extended inverse Weibull distribution with reliability application.” Journal of the Egyptian Mathematical Society, 25, 343–349. doi:10.1016/j.joems.2017.02.006, http://dx.doi.org/10.1016/j.joems.2017.02.006.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rMOEIW(n=400, mu=0.6, sigma=1.7, nu=0.3)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='MOEIW',
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 <- 400
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-2.02 + 3 * x1)
sigma <- exp(2.23 - 2 * x2)
nu <- 0.3
x <- rMOEIW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=MOEIW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Marshall-Olkin Extended Weibull family
Description
The Marshall-Olkin Extended Weibull family
Usage
MOEW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Marshall-Olkin Extended Weibull distribution with parameters mu,
sigma and nu has density given by
f(x) = \frac{\mu \sigma \nu (\nu x)^{\sigma - 1} exp\{{-(\nu x )^{\sigma}}\}}{\{1-(1-\mu) exp\{{-(\nu x )^{\sigma}}\} \}^{2}},
for x > 0.
Value
Returns a gamlss.family object which can be used to fit a MOEW distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
M.E G, E.K A, R.A J (2005). “Marshall–Olkin extended weibull distribution and its application to censored data.” Journal of Applied Statistics, 32(10), 1025–1034. doi:10.1080/02664760500165008.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rMOEW(n=400, mu=0.5, sigma=0.7, nu=1)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='MOEW',
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 <- 500
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-1.20 + 3 * x1)
sigma <- exp(0.84 - 2 * x2)
nu <- 1
x <- rMOEW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=MOEW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Marshall-Olkin Kappa family
Description
The Marshall-Olkin Kappa family
Usage
MOK(mu.link = "log", sigma.link = "log", nu.link = "log", tau.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
tau.link |
defines the tau.link, with "log" link as the default for the tau parameter. |
Details
The Marshall-Olkin Kappa distribution with parameters mu,
sigma, nu and tau has density given by
f(x)=\frac{\tau\frac{\mu\nu}{\sigma}\left(\frac{x}{\sigma}\right)^{\nu-1} \left(\mu+\left(\frac{x}{\sigma}\right)^{\mu\nu}\right)^{-\frac{\mu+1}{\mu}}}{\left(\tau+(1-\tau)\left(\frac{\left(\frac{x}{\sigma}\right)^{\mu\nu}}{\mu+\left(\frac{x}{\sigma}\right)^{\mu\nu}}\right)^{\frac{1}{\mu}}\right)^2}
for x > 0.
Value
Returns a gamlss.family object which can be used to fit a MOK distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
References
Javed M, Nawaz T, Irfan M (2018). “The Marshall-Olkin kappa distribution: properties and applications.” Journal of King Saud University-Science.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rMOK(n=100, mu = 1, sigma = 3.5, nu = 3, tau = 2)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, tau.fo=~1, family=MOK,
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 <- 0.5
x <- rMOK(n=n, mu, sigma, nu, tau)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~1, family=MOK,
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"))
The Modified Weibull family
Description
#' The Modified Weibull distribution
Usage
MW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Modified Weibull distribution with parameters mu,
sigma and nu has density given by
f(x) = \mu (\sigma + \nu x) x^(\sigma - 1) \exp(\nu x) \exp(-\mu x^(\sigma) \exp(\nu x)),
for x > 0, \mu > 0, \sigma \geq 0 and \nu \geq 0.
Value
Returns a gamlss.family object which can be used to fit a MW distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
Lai CD, Xie M, Murthy DNP (2003). “A modified Weibull distribution.” IEEE Transactions on reliability, 52(1), 33–37.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rMW(n=100, mu = 2, sigma = 1.5, nu = 0.2)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family= 'MW',
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 <- rpois(n, lambda=2)
x2 <- runif(n)
mu <- exp(3 -1 * x1)
sigma <- exp(2 - 2 * x2)
nu <- 0.2
x <- rMW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, mu.fo=~x1, sigma.fo=~x2, nu.fo=~1, family=MW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
coef(mod, what='nu')
The Odd Weibull family
Description
The function OW() defines the Odd Weibull distribution, a three parameter
distribution, for a gamlss.family object to be used in GAMLSS fitting
using the function gamlss().
Usage
OW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu. |
Details
The odd Weibull with parameters mu, sigma and nu
has density given by
f(t) = \left( \frac{\sigma\nu}{t} \right) (\mu t)^\sigma
e^{(\mu t)^\sigma} \left(e^{(\mu t)^{\sigma}}-1\right)^{\nu-1}
\left[ 1 + \left(e^{(\mu t)^{\sigma}}-1\right)^\nu \right]^{-2}
for x > 0.
Value
Returns a gamlss.family object which can be used to fit a OW distribution in the gamlss() function.
Author(s)
Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co
References
Cooray K (2006). “Generalization of the Weibull distribution: The odd Weibull family.” Statistical Modelling, 6(3), 265–277. ISSN 1471082X, doi:10.1191/1471082X06st116oa.
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rOW(n=200, mu=0.1, sigma=7, nu = 1.1)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family="OW",
control=gamlss.control(n.cyc=500, 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 <- 500
x1 <- runif(n)
x2 <- runif(n)
x3 <- runif(n)
mu <- exp(1.2 + 2 * x1)
sigma <- 2.12 + 3 * x2
nu <- exp(0.2 - x3)
x <- rOW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~x3,
family=OW(sigma.link='identity'),
control=gamlss.control(n.cyc=300, trace=FALSE))
coef(mod, what='mu')
coef(mod, what='sigma')
coef(mod, what='nu')
The Power Lindley family
Description
Power Lindley distribution
Usage
PL(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 Power Lindley Distribution with parameters mu
and sigma has density given by
f(x) = \frac{\mu \sigma^2}{\sigma + 1} (1 + x^\mu) x ^ {\mu - 1} \exp({-\sigma x ^\mu}),
for x > 0.
Value
Returns a gamlss.family object which can be used to fit a PL distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
Ghitanya ME, Al-Mutairi DK, Balakrishnanb N, Al-Enezi LJ (2013). “Power Lindley distribution and associated inference.” Computational Statistics and Data Analysis, 64, 20–33. doi:10.1016/j.csda.2013.02.026.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rPL(n=100, mu=1.5, sigma=0.2)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, family= 'PL',
control=gamlss.control(n.cyc=5000, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod, 'mu'))
exp(coef(mod, 'sigma'))
# 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(1.2 - 2 * x1)
sigma <- exp(0.8 - 3 * x2)
x <- rPL(n=n, mu, sigma)
mod <- gamlss(x~x1, sigma.fo=~x2, family=PL,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
The Quasi XGamma Poisson family
Description
The Quasi XGamma Poisson family
Usage
QXGP(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Quasi XGamma Poisson distribution with parameters mu,
sigma and nu has density given by
f(x)= K(\mu, \sigma, \nu)(\frac {\sigma^{2} x^{2}}{2} + \mu)
exp(\frac{\nu exp(-\sigma x)(1 + \mu + \sigma x + \frac {\sigma^{2}x^{2}}{2})}{1+\mu} - \sigma x),
for x > 0, \mu> 0, \sigma> 0, \nu> 1.
where
K(\mu, \sigma, \nu) = \frac{\nu \sigma}{(exp(\nu)-1)(1+\mu)}
Value
Returns a gamlss.family object which can be used to fit a QXGP distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
References
Subhradev S, Mustafa C K, Haitham M Y (2018). “The Quasi XGamma-Poisson distribution: Properties and Application.” Istatistik: Journal of the Turkish Statistical Assocation, 11(3), 65–76. ISSN 1300-4077, https://dergipark.org.tr/en/pub/ijtsa/issue/42850/518206.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rQXGP(n=200, mu=4, sigma=2, nu=3)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='QXGP',
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 <- 2000
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-2.19 + 3 * x1)
sigma <- exp(1 - 2 * x2)
nu <- 1
x <- rQXGP(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=QXGP,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Reflected Weibull family
Description
Reflected Weibull distribution
Usage
RW(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 Reflected Weibull Distribution with parameters mu
and sigma has density given by
f(y) = \mu\sigma (-y) ^{\sigma - 1} e ^ {-\mu(-y)^\sigma},
for y < 0
Value
Returns a gamlss.family object which can be used to fit a RW distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
Clifford Cohen A (1973). “The Reflected Weibull Distribution.” Technometrics, 15(4), 867–873. doi:10.2307/1267396.
See Also
Examples
# Example 1
# Generating some random values with
# known mu and sigma
y <- rRW(n=100, mu=1, sigma=1)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, family= 'RW',
control=gamlss.control(n.cyc=5000, trace=FALSE))
# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod, 'mu'))
exp(coef(mod, 'sigma'))
# 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(1.5 - 1.5 * x1)
sigma <- exp(2 - 2 * x2)
x <- rRW(n=n, mu, sigma)
mod <- gamlss(x~x1, sigma.fo=~x2, family=RW,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
The Sarhan and Zaindin's Modified Weibull family
Description
The Sarhan and Zaindin's Modified Weibull distribution
Usage
SZMW(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Sarhan and Zaindin's Modified Weibull distribution with parameters mu,
sigma and nu has density given by
f(x)=(\mu + \sigma \nu x^(\nu - 1)) \exp(- \mu x - \sigma x^\nu),
for x > 0, \mu > 0, \sigma > 0 and \nu > 0.
Value
Returns a gamlss.family object which can be used to fit a SZMW distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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, Zaindin M (2009). “Modified Weibull distribution.” APPS. Applied Sciences, 11, 123–136.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rSZMW(n=100, mu = 1, sigma = 1, nu = 1.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='SZMW',
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)
x2 <- runif(n)
mu <- exp(-1.6 * x1)
sigma <- exp(0.9 - 1 * x2)
nu <- 1.5
x <- rSZMW(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, mu.fo=~x1, sigma.fo=~x2, nu.fo=~1, family=SZMW,
control=gamlss.control(n.cyc=50000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
coef(mod, what='nu')
The Weibull Geometric family
Description
The Weibull Geometric distribution
Usage
WG(mu.link = "log", sigma.link = "log", nu.link = "logit")
Arguments
mu.link |
defines the mu.link, with "log" link as the default for the mu parameter. |
sigma.link |
defines the sigma.link, with "log" link as the default for the sigma. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The weibull geometric distribution with parameters mu,
sigma and nu has density given by
f(x) = (\sigma \mu^\sigma (1-\nu) x^(\sigma - 1) \exp(-(\mu x)^\sigma))
(1- \nu \exp(-(\mu x)^\sigma))^{-2},
for x > 0, \mu > 0, \sigma > 0 and 0 < \nu < 1.
Value
Returns a gamlss.family object which can be used to fit a WG distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
References
Barreto-Souza W, de Morais AL, Cordeiro GM (2011). “The Weibull-geometric distribution.” Journal of Statistical Computation and Simulation, 81(5), 645–657.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rWG(n=100, mu = 0.9, sigma = 2, nu = 0.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='WG',
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)
x2 <- runif(n)
mu <- exp(- 0.2 * x1)
sigma <- exp(1.2 - 1 * x2)
nu <- 0.5
x <- rWG(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, mu.fo=~x1, sigma.fo=~x2, nu.fo=~1, family=WG,
control=gamlss.control(n.cyc=50000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
coef(mod, what='nu')
The Weigted Generalized Exponential-Exponential family
Description
The Weigted Generalized Exponential-Exponential family
Usage
WGEE(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Weigted Generalized Exponential-Exponential distribution with parameters mu,
sigma and nu has density given by
f(x)= \sigma \nu \exp(-\nu x) (1 - \exp(-\nu x))^{\sigma - 1} (1 - \exp(-\mu \nu x)) / 1 - \sigma B(\mu + 1, \sigma),
for x > 0, \mu > 0, \sigma > 0 and \nu > 0.
Value
Returns a gamlss.family object which can be used to fit a WGEE distribution in the gamlss() function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
References
Mahdavi A (2015). “Two weighted distributions generated by exponential distribution.” Journal of Mathematical Extension, 9, 1–12.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rWGEE(n=1000, mu = 5, sigma = 0.5, nu = 1)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='WGEE',
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 <- 500
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(2 - x1)
sigma <- exp(1 - 3*x2)
nu <- 1
x <- rWGEE(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=WGEE,
control=gamlss.control(n.cyc=50000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Weibull Poisson family
Description
The Weibull Poisson family
Usage
WP(mu.link = "log", sigma.link = "log", nu.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. |
nu.link |
defines the nu.link, with "log" link as the default for the nu parameter. |
Details
The Weibull Poisson distribution with parameters mu,
sigma and nu has density given by
f(x) = \frac{\mu \sigma \nu e^{-\nu}} {1-e^{-\nu}} x^{\mu-1} exp({-\sigma x^{\mu}+\nu exp({-\sigma} x^{\mu}) }),
for x > 0.
Value
Returns a gamlss.family object which can be used to fit a WP distribution in the gamlss() function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
Wanbo L, Daimin S (1967). “A new compounding life distribution: the Weibull–Poisson distribution.” Journal of Applied Statistics, 9(1), 21–38. doi:10.1080/02664763.2011.575126, https://doi.org/10.1080/02664763.2011.575126.
See Also
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rWP(n=300, mu=1.5, sigma=0.5, nu=0.5)
# Fitting the model
require(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, family='WP',
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 <- 2000
x1 <- runif(n, min=0.4, max=0.6)
x2 <- runif(n, min=0.4, max=0.6)
mu <- exp(-1.3 + 3 * x1)
sigma <- exp(0.69 - 2 * x2)
nu <- 0.5
x <- rWP(n=n, mu, sigma, nu)
mod <- gamlss(x~x1, sigma.fo=~x2, nu.fo=~1, family=WP,
control=gamlss.control(n.cyc=5000, trace=FALSE))
coef(mod, what="mu")
coef(mod, what="sigma")
exp(coef(mod, what="nu"))
The Additive Weibull distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
nu |
shape parameter. |
tau |
shape 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 observations. |
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, amylkar.urrea@udea.edu.co
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
The Beta Generalized Exponentiated distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
nu |
parameter. |
tau |
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 observations. |
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, johand.marin@udea.edu.co
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
The Cosine Sine Exponential distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
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
The Extended Exponential Geometric distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
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 observations. |
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, johand.marin@udea.edu.co
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
The four parameter Exponentiated Generalized Gamma distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
nu |
parameter. |
tau |
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 observations. |
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, amylkar.urrea@udea.edu.co
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
The Exponentiated Modifien Weibull Extension distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
nu |
parameter. |
tau |
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 observations. |
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, johand.marin@udea.edu.co
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
The Extended Odd Frechet-Nadarajah-Haghighi
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
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
nu |
parameter. |
tau |
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 observations. |
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
The Exponentiated Weibull distribution
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
x, q |
vector of quantiles. |
mu |
scale parameter. |
sigma, nu |
shape parameters. |
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 observations. |
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
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
The Extended Weibull distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
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, amylkar.urrea@udea.edu.co
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
The Flexible Weibull Extension distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
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 observations. |
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
The Generalized Gompertz distribution
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
x, q |
vector of quantiles. |
mu, nu |
scale parameter. |
sigma |
shape parameters. |
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 observations. |
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, johand.marin@udea.edu.co
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
The Generalized Inverse Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Generalized Inverse Weibull distribution
with parameters mu, sigma and nu.
Usage
dGIW(x, mu, sigma, nu, log = FALSE)
pGIW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qGIW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rGIW(n, mu, sigma, nu)
hGIW(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
Details
The Generalized Inverse Weibull distribution mu,
sigma and nu has density given by
f(x) = \nu \sigma \mu^{\sigma} x^{-(\sigma + 1)} exp \{-\nu (\frac{\mu}{x})^{\sigma}\},
for x > 0.
Value
dGIW gives the density, pGIW gives the distribution
function, qGIW gives the quantile function, rGIW
generates random deviates and hGIW gives the hazard function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
Felipe R SdG, Edwin M MO, Gauss M C (2009). “The generalized inverse Weibull distribution.” Statistical papers, 52(3), 591–619. doi:10.1007/s00362-009-0271-3.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dGIW(x, mu=3, sigma=5, nu=0.5), from=0.001, to=8,
col="red", ylab="f(x)", las=1)
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pGIW(x, mu=3, sigma=5, nu=0.5),
from=0.0001, to=14, col="red", las=1, ylab="F(x)")
curve(pGIW(x, mu=3, sigma=5, nu=0.5, lower.tail=FALSE),
from=0.0001, to=14, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qGIW(p, mu=3, sigma=5, nu=0.5), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pGIW(x, mu=3, sigma=5, nu=0.5),
from=0, add=TRUE, col="red")
## The random function
hist(rGIW(n=1000, mu=3, sigma=5, nu=0.5), freq=FALSE,
xlab="x", ylim=c(0, 0.8), las=1, main="")
curve(dGIW(x, mu=3, sigma=5, nu=0.5),
from=0.001, to=14, add=TRUE, col="red")
## The Hazard function
curve(hGIW(x, mu=3, sigma=5, nu=0.5), from=0.001, to=30,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Generalized modified Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the generalized
modified weibull distribution with parameters mu,
sigma, nu and tau.
Usage
dGMW(x, mu, sigma, nu, tau, log = FALSE)
pGMW(q, mu, sigma, nu, tau, lower.tail = TRUE, log.p = FALSE)
qGMW(p, mu, sigma, nu, tau, lower.tail = TRUE, log.p = FALSE)
rGMW(n, mu, sigma, nu, tau)
hGMW(x, mu, sigma, nu, tau, log = FALSE)
Arguments
x, q |
vector of quantiles. |
mu |
scale parameter. |
sigma |
shape parameter. |
nu |
shape parameter. |
tau |
acceleration 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 observations. |
Details
The generalized modified weibull with parameters mu,
sigma, nu and tau has density given by
f(x)= \mu \sigma x^{\nu - 1}(\nu + \tau x) \exp(\tau x - \mu x^{\nu} e^{\tau x})
[1 - \exp(- \mu x^{\nu} e^{\tau x})]^{\sigma-1},
for x>0.
Value
dGMW gives the density, pGMW gives the distribution
function, qGMW gives the quantile function, rGMW
generates random deviates and hGMW gives the hazard function.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, to=0.8,
ylim=c(0, 2), col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5),
from=0, to=1.2, col="red", las=1, ylab="F(x)")
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5, lower.tail=FALSE),
from=0, to=1.2, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qGMW(p, mu=2, sigma=0.5, nu=2, tau=1.5), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, add=TRUE, col="red")
## The random function
hist(rGMW(n=1000, mu=2, sigma=0.5, nu=2,tau=1.5), freq=FALSE,
xlab="x", main ="", las=1)
curve(dGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hGMW(x, mu=2, sigma=0.5, nu=2, tau=1.5), from=0, to=1, ylim=c(0, 16),
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Gamma Weibull distribution
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
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
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, johand.marin@udea.edu.co
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
The Inverse Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the inverse weibull distribution with
parameters mu and sigma.
Usage
dIW(x, mu, sigma, log = FALSE)
pIW(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qIW(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rIW(n, mu, sigma)
hIW(x, mu, sigma)
Arguments
x, q |
vector of quantiles. |
mu |
scale parameter. |
sigma |
shape parameters. |
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 observations. |
Details
The inverse weibull distribution with parameters mu and
sigma has density given by
f(x) = \mu \sigma x^{-\sigma-1} \exp(\mu x^{-\sigma})
for x > 0, \mu > 0 and \sigma > 0
Value
dIW gives the density, pIW gives the distribution
function, qIW gives the quantile function, rIW
generates random deviates and hIW gives the hazard function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
Drapella A (1993). “The complementary Weibull distribution: unknown or just forgotten?” Quality and Reliability Engineering International, 9(4), 383–385.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dIW(x, mu=5, sigma=2.5), from=0, to=10,
ylim=c(0, 0.55), col="red", las=1, ylab="f(x)")
#'
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pIW(x, mu=5, sigma=2.5),
from=0, to=10, col="red", las=1, ylab="F(x)")
curve(pIW(x, mu=5, sigma=2.5, lower.tail=FALSE),
from=0, to=10, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qIW(p, mu=5, sigma=2.5), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pIW(x, mu=5, sigma=2.5), from=0, add=TRUE, col="red")
## The random function
hist(rIW(n=10000, mu=5, sigma=2.5), freq=FALSE, xlim=c(0,60),
xlab="x", las=1, main="")
curve(dIW(x, mu=5, sigma=2.5), from=0, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hIW(x, mu=5, sigma=2.5), from=0, to=15, ylim=c(0, 0.9),
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Kumaraswamy Inverse Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Kumaraswamy Inverse Weibull distribution
with parameters mu, sigma and nu.
Usage
dKumIW(x, mu, sigma, nu, log = FALSE)
pKumIW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qKumIW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rKumIW(n, mu, sigma, nu)
hKumIW(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
Details
The Kumaraswamy Inverse Weibull Distribution with parameters mu,
sigma and nu has density given by
f(x)= \mu \sigma \nu x^{-\mu - 1} \exp{- \sigma x^{-\mu}} (1 - \exp{- \sigma x^{-\mu}})^{\nu - 1},
for x > 0, \mu > 0, \sigma > 0 and \nu > 0.
Value
dKumIW gives the density, pKumIW gives the distribution
function, qKumIW gives the quantile function, rKumIW
generates random deviates and hKumIW gives the hazard function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
Shahbaz MQ, Shahbaz S, Butt NS (2012). “The Kumaraswamy-Inverse Weibull Distribution.” Shahbaz, MQ, Shahbaz, S., & Butt, NS (2012). The Kumaraswamy–Inverse Weibull Distribution. Pakistan journal of statistics and operation research, 8(3), 479–489.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
par(mfrow = c(1, 1))
curve(dKumIW(x, mu = 1.5, sigma= 1.5, nu = 1), from = 0, to = 8.5,
col = "red", las = 1, ylab = "f(x)")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pKumIW(x, mu = 1.5, sigma= 1.5, nu = 1), from = 0, to = 8.5,
ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pKumIW(x, mu = 1.5, sigma= 1.5, nu = 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 = qKumIW(p=p, mu = 1.5, sigma= 1.5, nu = 10), y = p,
xlab = "Quantile", las = 1, ylab = "Probability")
curve(pKumIW(x, mu = 1.5, sigma= 1.5, nu = 10), from = 0, add = TRUE,
col = "red")
## The random function
hist(rKumIW(1000, mu = 1.5, sigma= 1.5, nu = 5), freq = FALSE, xlab = "x",
las = 1, ylim = c(0, 1.5), main = "")
curve(dKumIW(x, mu = 1.5, sigma= 1.5, nu = 5), from = 0, to =8, add = TRUE,
col = "red")
## The Hazard function
par(mfrow=c(1,1))
curve(hKumIW(x, mu = 1.5, sigma= 1.5, nu = 1), from = 0, to = 3,
ylim = c(0, 0.7), col = "red", ylab = "Hazard function", las = 1)
par(old_par) # restore previous graphical parameters
Lindley distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Lindley distribution
with parameter mu.
Usage
dLIN(x, mu, log = FALSE)
pLIN(q, mu, lower.tail = TRUE, log.p = FALSE)
qLIN(p, mu, lower.tail = TRUE, log.p = FALSE)
rLIN(n, mu)
hLIN(x, mu, log = FALSE)
Arguments
x, q |
vector of quantiles. |
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 observations. |
Details
Lindley Distribution with parameter mu has density given by
f(x) = \frac{\mu^2}{\mu+1} (1+x) \exp(-\mu x),
for x > 0 and \mu > 0. These function were taken form LindleyR package.
Value
dLIN gives the density, pLIN gives the distribution
function, qLIN gives the quantile function, rLIN
generates random deviates and hLIN gives the hazard function.
Author(s)
Freddy Hernandez, fhernanb@unal.edu.co
References
Lindley DV (1958). “Fiducial distributions and Bayes' theorem.” Journal of the Royal Statistical Society. Series B (Methodological), 102–107.
Lindley DV (1965). Introduction to probability and statistics: from a Bayesian viewpoint. 2. Inference. CUP Archive.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dLIN(x, mu=1.5), from=0.0001, to=10,
col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pLIN(x, mu=2), from=0.0001, to=10, col="red", las=1, ylab="F(x)")
curve(pLIN(x, mu=2, lower.tail=FALSE), from=0.0001,
to=10, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qLIN(p, mu=2), y=p, xlab="Quantile", las=1, ylab="Probability")
curve(pLIN(x, mu=2), from=0, add=TRUE, col="red")
## The random function
hist(rLIN(n=10000, mu=2), freq=FALSE, xlab="x", las=1, main="")
curve(dLIN(x, mu=2), from=0.09, to=5, add=TRUE, col="red")
## The Hazard function
curve(hLIN(x, mu=2), from=0.001, to=10, col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Log-Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Log-Weibull distribution
with parameters mu and sigma.
Usage
dLW(x, mu, sigma, log = FALSE)
pLW(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qLW(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rLW(n, mu, sigma)
hLW(x, mu, sigma)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
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 observations. |
Details
The Log-Weibull Distribution with parameters mu
and sigma has density given by
f(y)=(1/\sigma) e^{((y - \mu)/\sigma)} exp\{-e^{((y - \mu)/\sigma)}\},
for - infty < y < infty.
Value
dLW gives the density, pLW gives the distribution
function, qLW gives the quantile function, rLW
generates random deviates and hLW gives the hazard function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
E.J G (1958). Statistics of extremes. Columbia University Press. ISBN 10:0231021909.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dLW(x, mu=0, sigma=1.5), from=-8, to=5,
col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pLW(x, mu=0, sigma=1.5),
from=-8, to=5, col="red", las=1, ylab="F(x)")
curve(pLW(x, mu=0, sigma=1.5, lower.tail=FALSE),
from=-8, to=5, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qLW(p, mu=0, sigma=1.5), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pLW(x, mu=0, sigma=1.5), from=-8, to=5, add=TRUE, col="red")
## The random function
hist(rLW(n=10000, mu=0, sigma=1.5), freq=FALSE,
xlab="x", las=1, main="")
curve(dLW(x, mu=0, sigma=1.5), from=-15, to=6, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hLW(x, mu=0, sigma=1.5), from=-8, to=7,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Marshall-Olkin Extended Inverse Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Marshall-Olkin Extended Inverse Weibull distribution
with parameters mu, sigma and nu.
Usage
dMOEIW(x, mu, sigma, nu, log = FALSE)
pMOEIW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qMOEIW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rMOEIW(n, mu, sigma, nu)
hMOEIW(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
Details
The Marshall-Olkin Extended Inverse Weibull distribution mu,
sigma and nu has density given by
f(x) = \frac{\mu \sigma \nu x^{-(\sigma + 1)} exp\{{-\mu x^{-\sigma}}\}}{\{\nu -(\nu-1) exp\{{-\mu x ^{-\sigma}}\} \}^{2}},
for x > 0.
Value
dMOEIW gives the density, pMOEIW gives the distribution
function, qMOEIW gives the quantile function, rMOEIW
generates random deviates and hMOEIW gives the hazard function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
References
Hassan M O, A.H E, A.M.K T, Abdulkareem M Bc (2017). “Extended inverse Weibull distribution with reliability application.” Journal of the Egyptian Mathematical Society, 25, 343–349. doi:10.1016/j.joems.2017.02.006, http://dx.doi.org/10.1016/j.joems.2017.02.006.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dMOEIW(x, mu=0.6, sigma=1.7, nu=0.3), from=0, to=2,
col="red", ylab="f(x)", las=1)
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pMOEIW(x, mu=0.6, sigma=1.7, nu=0.3),
from=0.0001, to=2, col="red", las=1, ylab="F(x)")
curve(pMOEIW(x, mu=0.6, sigma=1.7, nu=0.3, 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=qMOEIW(p, mu=0.6, sigma=1.7, nu=0.3), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pMOEIW(x, mu=0.6, sigma=1.7, nu=0.3),
from=0, add=TRUE, col="red")
## The random function
hist(rMOEIW(n=1000, mu=0.6, sigma=1.7, nu=0.3), freq=FALSE,
xlab="x", las=1, main="")
curve(dMOEIW(x, mu=0.6, sigma=1.7, nu=0.3),
from=0.001, to=4, add=TRUE, col="red")
## The Hazard function
curve(hMOEIW(x, mu=0.5, sigma=0.7, nu=1), from=0.001, to=3,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Marshall-Olkin Extended Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Marshall-Olkin Extended Weibull distribution
with parameters mu, sigma and nu.
Usage
dMOEW(x, mu, sigma, nu, log = FALSE)
pMOEW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qMOEW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rMOEW(n, mu, sigma, nu)
hMOEW(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
Details
The Marshall-Olkin Extended Weibull distribution mu,
sigma and nu has density given by
f(x) = \frac{\mu \sigma \nu (\nu x)^{\sigma - 1} exp\{{-(\nu x )^{\sigma}}\}}{\{1-(1-\mu) exp\{{-(\nu x )^{\sigma}}\} \}^{2}},
for x > 0.
Value
dMOEW gives the density, pMOEW gives the distribution
function, qMOEW gives the quantile function, rMOEW
generates random deviates and hMOEW gives the hazard function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
M.E G, E.K A, R.A J (2005). “Marshall–Olkin extended weibull distribution and its application to censored data.” Journal of Applied Statistics, 32(10), 1025–1034. doi:10.1080/02664760500165008.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dMOEW(x, mu=0.5, sigma=0.7, nu=1), from=0.001, to=1,
col="red", ylab="f(x)", las=1)
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pMOEW(x, mu=0.5, sigma=0.7, nu=1),
from=0.0001, to=2, col="red", las=1, ylab="F(x)")
curve(pMOEW(x, mu=0.5, sigma=0.7, nu=1, 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=qMOEW(p, mu=0.5, sigma=0.7, nu=1), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pMOEW(x, mu=0.5, sigma=0.7, nu=1),
from=0, add=TRUE, col="red")
## The random function
hist(rMOEW(n=100, mu=0.5, sigma=0.7, nu=1), freq=FALSE,
xlab="x", ylim=c(0, 1), las=1, main="")
curve(dMOEW(x, mu=0.5, sigma=0.7, nu=1),
from=0.001, to=2, add=TRUE, col="red")
## The Hazard function
curve(hMOEW(x, mu=0.5, sigma=0.7, nu=1), from=0.001, to=3,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Marshall-Olkin Kappa distribution
Description
Desnsity, distribution function, quantile function,
random generation and hazard function for the Marshall-Olkin Kappa distribution
with parameters mu, sigma, nu and tau.
Usage
dMOK(x, mu, sigma, nu, tau, log = FALSE)
pMOK(q, mu, sigma, nu, tau, lower.tail = TRUE, log.p = FALSE)
qMOK(p, mu, sigma, nu, tau, lower.tail = TRUE, log.p = FALSE)
rMOK(n, mu, sigma, nu, tau)
hMOK(x, mu, sigma, nu, tau)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
nu |
parameter. |
tau |
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 observations. |
Details
The Marshall-Olkin Kappa distribution with parameters mu,
sigma, nu and tau has density given by:
f(x)=\frac{\tau\frac{\mu\nu}{\sigma}\left(\frac{x}{\sigma}\right)^{\nu-1} \left(\mu+\left(\frac{x}{\sigma}\right)^{\mu\nu}\right)^{-\frac{\mu+1}{\mu}}}{\left[\tau+(1-\tau)\left(\frac{\left(\frac{x}{\sigma}\right)^{\mu\nu}}{\mu+\left(\frac{x}{\sigma}\right)^{\mu\nu}}\right)^{\frac{1}{\mu}}\right]^2}
for x > 0.
Value
dMOK gives the density, pMOK gives the distribution function,
qMOK gives the quantile function, rMOK generates random deviates
and hMOK gives the hazard function.
Author(s)
Angel Muñoz,
References
Javed M, Nawaz T, Irfan M (2018). “The Marshall-Olkin kappa distribution: properties and applications.” Journal of King Saud University-Science.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
par(mfrow = c(1,1))
curve(dMOK(x = x, mu = 1, sigma = 3.5, nu = 3, tau = 2), from = 0, to = 15,
ylab = 'f(x)', col = 2, las = 1)
## The cumulative distribution and the Reliability function
par(mfrow = c(1,2))
curve(pMOK(q = x, mu = 1, sigma = 2.5, nu = 3, tau = 2), from = 0, to = 10,
col = 2, lwd = 2, las = 1, ylab = 'F(x)')
curve(pMOK(q = x, mu = 1, sigma = 2.5, nu = 3, tau = 2, lower.tail = FALSE), from = 0, to = 10,
col = 2, lwd = 2, las = 1, ylab = 'R(x)')
## The quantile function
p <- seq(from = 0.00001, to = 0.99999, length.out = 100)
plot(x = qMOK(p = p, mu = 4, sigma = 2.5, nu = 3, tau = 2), y = p, xlab = 'Quantile',
las = 1, ylab = 'Probability')
curve(pMOK(q = x, mu = 4, sigma = 2.5, nu = 3, tau = 2), from = 0, to = 15,
add = TRUE, col = 2)
## The random function
hist(rMOK(n = 10000, mu = 1, sigma = 2.5, nu = 3, tau = 2), freq = FALSE,
xlab = "x", las = 1, main = '', ylim = c(0,.3), xlim = c(0,20), breaks = 50)
curve(dMOK(x, mu = 1, sigma = 2.5, nu = 3, tau = 2), from = 0, to = 15, add = TRUE, col = 2)
## The Hazard function
par(mfrow = c(1,1))
curve(hMOK(x = x, mu = 1, sigma = 2.5, nu = 3, tau = 2), from = 0, to = 20,
col = 2, ylab = 'Hazard function', las = 1)
par(old_par) # restore previous graphical parameters
The Modified Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the modified weibull distribution
with parameters mu, sigma and nu.
Usage
dMW(x, mu, sigma, nu, log = FALSE)
pMW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qMW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rMW(n, mu, sigma, nu)
hMW(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
shape parameter one. |
sigma |
parameter two. |
nu |
scale parameter three. |
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 observations. |
Details
The modified weibull distribution with parameters mu, sigma
and nu has density given by
f(x) = \mu (\sigma + \nu x) x^{\sigma - 1} \exp(\nu x) \exp(-\mu x^{\sigma} \exp(\nu x))
for x > 0, \mu > 0, \sigma \geq 0 and \nu \geq 0.
Value
dMW gives the density, pMW gives the distribution
function, qMW gives the quantile function, rMW
generates random deviates and hMW gives the hazard function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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.
Lai CD, Xie M, Murthy DNP (2003). “A modified Weibull distribution.” IEEE Transactions on reliability, 52(1), 33–37.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=2,
ylim=c(0, 1.5), col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=2,
col = "red", las=1, ylab="F(x)")
curve(pMW(x, mu=2, sigma=1.5, nu=0.2, lower.tail = FALSE),
from=0, to=2, col="red", las=1, ylab ="R(x)")
## The quantile function
p <- seq(from=0, to=0.9999, length.out=100)
plot(x=qMW(p, mu=2, sigma=1.5, nu=0.2), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pMW(x, mu=2, sigma=1.5, nu=0.2), from=0, add=TRUE, col="red")
## The random function
hist(rMW(n=1000, mu=2, sigma=1.5, nu=0.2), freq=FALSE,
xlab="x", las=1, main="")
curve(dMW(x, mu=2, sigma=1.5, nu=0.2), from=0, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hMW(x, mu=2, sigma=1.5, nu=0.2), from=0, to=1.5, ylim=c(0, 5),
col="red", las=1, ylab="H(x)", las=1)
par(old_par) # restore previous graphical parameters
The Odd Weibull Distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Odd Weibull distribution with
parameters mu, sigma and nu.
Usage
dOW(x, mu, sigma, nu, log = FALSE)
pOW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qOW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rOW(n, mu, sigma, nu)
hOW(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
parameter one. |
sigma |
parameter two. |
nu |
parameter three. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[T <= t], otherwise, P[T > t]. |
p |
vector of probabilities. |
n |
number of observations. |
Details
The Odd Weibull with parameters mu, sigma
and nu has density given by
f(x) = \left( \frac{\sigma\nu}{x} \right) (\mu x)^\sigma
e^{(\mu x)^\sigma} \left(e^{(\mu x)^{\sigma}}-1\right)^{\nu-1}
\left[ 1 + \left(e^{(\mu x)^{\sigma}}-1\right)^\nu \right]^{-2}
for x > 0.
Value
dOW gives the density, pOW gives the distribution
function, qOW gives the quantile function, rOW
generates random deviates and hOW gives the hazard function.
Author(s)
Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co
References
Cooray K (2006). “Generalization of the Weibull distribution: The odd Weibull family.” Statistical Modelling, 6(3), 265–277. ISSN 1471082X, doi:10.1191/1471082X06st116oa.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dOW(x, mu=2, sigma=3, nu=0.2), from=0, to=4, ylim=c(0, 2),
col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pOW(x, mu=2, sigma=3, nu=0.2), from=0, to=4, ylim=c(0, 1),
col="red", las=1, ylab="f(x)")
curve(pOW(x, mu=2, sigma=3, nu=0.2, 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=0.998, length.out=100)
plot(x = qOW(p, mu=2, sigma=3, nu=0.2), y=p, xlab="Quantile", las=1,
ylab="Probability")
curve(pOW(x, mu=2, sigma=3, nu=0.2), from=0, add=TRUE, col="red")
## The random function
hist(rOW(n=10000, mu=2, sigma = 3, nu = 0.2), freq=FALSE, ylim = c(0, 2),
xlab = "x", las = 1, main = "")
curve(dOW(x, mu=2, sigma=3, nu=0.2), from=0, ylim=c(0, 2), add=TRUE,
col = "red")
## The Hazard function
par(mfrow=c(1,1))
curve(hOW(x, mu=2, sigma=3, nu=0.2), from=0, to=2.5, ylim=c(0, 3),
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Power Lindley distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Power Lindley distribution
with parameters mu and sigma.
Usage
dPL(x, mu, sigma, log = FALSE)
pPL(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qPL(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rPL(n, mu, sigma)
hPL(x, mu, sigma)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
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 observations. |
Details
The Power Lindley Distribution with parameters mu
and sigma has density given by
f(x) = \frac{\mu \sigma^2}{\sigma + 1} (1 + x^\mu) x ^ {\mu - 1} \exp({-\sigma x ^\mu}),
for x > 0.
Value
dPL gives the density, pPL gives the distribution
function, qPL gives the quantile function, rPL
generates random deviates and hPL gives the hazard function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
Ghitanya ME, Al-Mutairi DK, Balakrishnanb N, Al-Enezi LJ (2013). “Power Lindley distribution and associated inference.” Computational Statistics and Data Analysis, 64, 20–33. doi:10.1016/j.csda.2013.02.026.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dPL(x, mu=1.5, sigma=0.2), from=0.1, to=10,
col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pPL(x, mu=1.5, sigma=0.2),
from=0.1, to=10, col="red", las=1, ylab="F(x)")
curve(pPL(x, mu=1.5, sigma=0.2, lower.tail=FALSE),
from=0.1, to=10, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qPL(p, mu=1.5, sigma=0.2), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pPL(x, mu=1.5, sigma=0.2), from=0.1, add=TRUE, col="red")
## The random function
hist(rPL(n=1000, mu=1.5, sigma=0.2), freq=FALSE,
xlab="x", las=1, main="")
curve(dPL(x, mu=1.5, sigma=0.2), from=0.1, to=15, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hPL(x, mu=1.5, sigma=0.2), from=0.1, to=15,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Quasi XGamma Poisson distribution
Description
Density, distribution function,quantile function,
random generation and hazard function for the Quasi XGamma Poisson distribution
with parameters mu, sigma and nu.
Usage
dQXGP(x, mu, sigma, nu, log = FALSE)
pQXGP(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qQXGP(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rQXGP(n, mu, sigma, nu)
hQXGP(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
Details
The Quasi XGamma Poisson distribution with parameters mu,
sigma and nu has density given by:
f(x)= K(\mu, \sigma, \nu)(\frac {\sigma^{2} x^{2}}{2} + \mu)
exp(\frac{\nu exp(-\sigma x)(1 + \mu + \sigma x + \frac {\sigma^{2}x^{2}}{2})}{1+\mu} - \sigma x),
for x > 0, \mu> 0, \sigma> 0, \nu> 1.
where
K(\mu, \sigma, \nu) = \frac{\nu \sigma}{(exp(\nu)-1)(1+\mu)}
Value
dQXGP gives the density, pQXGP gives the distribution
function, qQXGP gives the quantile function, rQXGP
generates random deviates and hQXGP gives the hazard function.
Author(s)
Simon Zapata
References
Subhradev S, Mustafa C K, Haitham M Y (2018). “The Quasi XGamma-Poisson distribution: Properties and Application.” Istatistik: Journal of the Turkish Statistical Assocation, 11(3), 65–76. ISSN 1300-4077, https://dergipark.org.tr/en/pub/ijtsa/issue/42850/518206.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dQXGP(x, mu=0.5, sigma=1, nu=1), from=0.1, to=8,
ylim=c(0, 0.6), col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pQXGP(x, mu=0.5, sigma=1, nu=1),
from=0.1, to=8, col="red", las=1, ylab="F(x)")
curve(pQXGP(x, mu=0.5, sigma=1, nu=1, lower.tail=FALSE),
from=0.1, to=8, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qQXGP(p, mu=0.5, sigma=1, nu=1), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pQXGP(x, mu=0.5, sigma=1, nu=1),
from=0.1, add=TRUE, col="red")
## The random function
hist(rQXGP(n=1000, mu=0.5, sigma=1, nu=1), freq=FALSE,
xlab="x", ylim=c(0, 0.4), las=1, main="", xlim=c(0, 15))
curve(dQXGP(x, mu=0.5, sigma=1, nu=1),
from=0.001, to=500, add=TRUE, col="red")
## The Hazard function
curve(hQXGP(x, mu=0.5, sigma=1, nu=1), from=0.01, to=3,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Reflected Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Reflected Weibull Distribution
with parameters mu and sigma.
Usage
dRW(x, mu, sigma, log = FALSE)
pRW(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qRW(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rRW(n, mu, sigma)
hRW(x, mu, sigma)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
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 observations. |
Details
The Reflected Weibull Distribution with parameters mu
and sigma has density given by
f(y) = \mu\sigma (-y) ^{\sigma - 1} e ^ {-\mu(-y)^\sigma},
for y < 0.
Value
dRW gives the density, pRW gives the distribution
function, qRW gives the quantile function, rRW
generates random deviates and hRW gives the hazard function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
Clifford Cohen A (1973). “The Reflected Weibull Distribution.” Technometrics, 15(4), 867–873. doi:10.2307/1267396.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dRW(x, mu=1, sigma=1), from=-5, to=-0.01,
col="red", las=1, ylab="f(x)")
## The cumulative distribution and the Reliability function
par(mfrow=c(1, 2))
curve(pRW(x, mu=1, sigma=1),
from=-5, to=-0.01, col="red", las=1, ylab="F(x)")
curve(pRW(x, mu=1, sigma=1, lower.tail=FALSE),
from=-5, to=-0.01, col="red", las=1, ylab="R(x)")
## The quantile function
p <- seq(from=0, to=0.99999, length.out=100)
plot(x=qRW(p, mu=1, sigma=1), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pRW(x, mu=1, sigma=1), from=-5, add=TRUE, col="red")
## The random function
hist(rRW(n=10000, mu=1, sigma=1), freq=FALSE,
xlab="x", las=1, main="")
curve(dRW(x, mu=1, sigma=1), from=-5, to=-0.01, add=TRUE, col="red")
## The Hazard function
par(mfrow=c(1,1))
curve(hRW(x, mu=1, sigma=1), from=-0.3, to=-0.01,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
The Sarhan and Zaindin's Modified Weibull distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for Sarhan and Zaindins modified weibull distribution
with parameters mu, sigma and nu.
Usage
dSZMW(x, mu, sigma, nu, log = FALSE)
pSZMW(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qSZMW(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rSZMW(n, mu, sigma, nu)
hSZMW(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
scale parameter. |
sigma |
shape parameter. |
nu |
shape 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 observations. |
Details
The Sarhan and Zaindins modified weibull with parameters mu,
sigma and nu has density given by
f(x)=(\mu + \sigma \nu x^(\nu - 1)) \exp(- \mu x - \sigma x^\nu)
for x > 0, \mu > 0, \sigma > 0 and \nu > 0.
Value
dSZMW gives the density, pSZMW gives the distribution
function, qSZMW gives the quantile function, rSZMW
generates random deviates and hSZMW gives the hazard function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
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, Zaindin M (2009). “Modified Weibull distribution.” APPS. Applied Sciences, 11, 123–136.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, to = 2,
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(pSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, to = 2, ylim = c(0, 1),
col = "red", las = 1, ylab = "F(x)")
curve(pSZMW(x, mu = 2, sigma = 1.5, nu = 0.2, lower.tail = FALSE), from = 0,
to = 2, 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 = qSZMW(p = p, mu = 2, sigma = 1.5, nu = 0.2), y = p, xlab = "Quantile",
las = 1, ylab = "Probability")
curve(pSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, add = TRUE, col = "red")
## The random function
hist(rSZMW(n = 1000, mu = 2, sigma = 1.5, nu = 0.2), freq = FALSE, xlab = "x",
las = 1, main = "")
curve(dSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, add = TRUE, col = "red")
## The Hazard function
par(mfrow=c(1,1))
curve(hSZMW(x, mu = 2, sigma = 1.5, nu = 0.2), from = 0, to = 3, ylim = c(0, 8),
col = "red", ylab = "Hazard function", las = 1)
par(old_par) # restore previous graphical parameters
The Weibull Geometric distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the weibull geometric distribution with
parameters mu, sigma and nu.
Usage
dWG(x, mu, sigma, nu, log = FALSE)
pWG(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qWG(p, sigma, mu, nu, lower.tail = TRUE, log.p = FALSE)
rWG(n, mu, sigma, nu)
hWG(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
scale parameter. |
sigma |
shape parameter. |
nu |
parameter of geometric random variable. |
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 observations. |
Details
The Weibull geometric distribution with parameters mu,
sigma and nu has density given by
f(x) = (\sigma \mu^\sigma (1-\nu) x^(\sigma - 1) \exp(-(\mu x)^\sigma))
(1- \nu \exp(-(\mu x)^\sigma))^{-2},
for x > 0, \mu > 0, \sigma > 0 and 0 < \nu < 1.
Value
dWG gives the density, pWG gives the distribution
function, qWG gives the quantile function, rWG
generates random deviates and hWG gives the hazard function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
References
Barreto-Souza W, de Morais AL, Cordeiro GM (2011). “The Weibull-geometric distribution.” Journal of Statistical Computation and Simulation, 81(5), 645–657.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dWG(x, mu = 0.9, sigma = 2, nu = 0.5), from = 0, to = 3,
ylim = c(0, 1.1), col = "red", las = 1, ylab = "f(x)")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pWG(x, mu = 0.9, sigma = 2, nu = 0.5), from = 0, to = 3,
ylim = c(0, 1), col = "red", las = 1, ylab = "F(x)")
curve(pWG(x, mu = 0.9, sigma = 2, nu = 0.5, 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 = qWG(p = p, mu = 0.9, sigma = 2, nu = 0.5), y = p,
xlab = "Quantile", las = 1, ylab = "Probability")
curve(pWG(x,mu = 0.9, sigma = 2, nu = 0.5), from = 0, add = TRUE,
col = "red")
## The random function
hist(rWG(1000, mu = 0.9, sigma = 2, nu = 0.5), freq = FALSE, xlab = "x",
ylim = c(0, 1.8), las = 1, main = "")
curve(dWG(x, mu = 0.9, sigma = 2, nu = 0.5), from = 0, add = TRUE,
col = "red", ylim = c(0, 1.8))
## The Hazard function(
par(mfrow=c(1,1))
curve(hWG(x, mu = 0.9, sigma = 2, nu = 0.5), from = 0, to = 8,
ylim = c(0, 12), col = "red", ylab = "Hazard function", las = 1)
par(old_par) # restore previous graphical parameters
The Weighted Generalized Exponential-Exponential distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Weighted Generalized Exponential-Exponential distribution
with parameters mu, sigma and nu.
Usage
dWGEE(x, mu, sigma, nu, log = FALSE)
pWGEE(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qWGEE(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rWGEE(n, mu, sigma, nu)
hWGEE(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
Details
The Weighted Generalized Exponential-Exponential Distribution with parameters mu,
sigma and nu has density given by
f(x)= \sigma \nu \exp(-\nu x) (1 - \exp(-\nu x))^{\sigma - 1} (1 - \exp(-\mu \nu x)) / 1 - \sigma B(\mu + 1, \sigma),
for x > 0, \mu > 0, \sigma > 0 and \nu > 0.
Value
dWGEE gives the density, pWGEE gives the distribution
function, qWGEE gives the quantile function, rWGEE
generates random deviates and hWGEE gives the hazard function.
Author(s)
Johan David Marin Benjumea, johand.marin@udea.edu.co
References
Mahdavi A (2015). “Two Weighted Distributions Generated by Exponential Distribution.” Journal of Mathematical Extension, 9(1), 1–12.
Mahdavi A (2015). “Two weighted distributions generated by exponential distribution.” Journal of Mathematical Extension, 9, 1–12.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dWGEE(x, mu = 5, sigma = 0.5, nu = 1), from = 0, to = 6,
ylim = c(0, 1), col = "red", las = 1, ylab = "The probability density function")
## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(pWGEE(x, mu = 5, sigma = 0.5, nu = 1), from = 0, to = 6,
ylim = c(0, 1), col = "red", las = 1, ylab = "The cumulative distribution function")
curve(pWGEE(x, mu = 5, sigma = 0.5, nu = 1, lower.tail = FALSE),
from = 0, to = 6, ylim = c(0, 1), col = "red", las = 1, ylab = "The Reliability function")
## The quantile function
p <- seq(from = 0, to = 0.99999, length.out = 100)
plot(x = qWGEE(p = p, mu = 5, sigma = 0.5, nu = 1), y = p,
xlab = "Quantile", las = 1, ylab = "Probability")
curve(pWGEE(x, mu = 5, sigma = 0.5, nu = 1), from = 0, add = TRUE,
col = "red")
## The random function
hist(rWGEE(1000, mu = 5, sigma = 0.5, nu = 1), freq = FALSE, xlab = "x",
ylim = c(0, 1), las = 1, main = "")
curve(dWGEE(x, mu = 5, sigma = 0.5, nu = 1), from = 0, add = TRUE,
col = "red", ylim = c(0, 1))
## The Hazard function(
par(mfrow=c(1,1))
curve(hWGEE(x, mu = 5, sigma = 0.5, nu = 1), from = 0, to = 6,
ylim = c(0, 1.4), col = "red", ylab = "The hazard function", las = 1)
par(old_par) # restore previous graphical parameters
The Weibull Poisson distribution
Description
Density, distribution function, quantile function,
random generation and hazard function for the Weibull Poisson distribution
with parameters mu, sigma and nu.
Usage
dWP(x, mu, sigma, nu, log = FALSE)
pWP(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
qWP(p, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)
rWP(n, mu, sigma, nu)
hWP(x, mu, sigma, nu)
Arguments
x, q |
vector of quantiles. |
mu |
parameter. |
sigma |
parameter. |
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 observations. |
Details
The Weibull Poisson distribution with parameters mu,
sigma and nu has density given by
f(x) = \frac{\mu \sigma \nu e^{-\nu}} {1-e^{-\nu}} x^{\mu-1} exp({-\sigma x^{\mu}+\nu exp({-\sigma} x^{\mu}) }),
for x > 0.
Value
dWP gives the density, pWP gives the distribution
function, qWP gives the quantile function, rWP
generates random deviates and hWP gives the hazard function.
Author(s)
Amylkar Urrea Montoya, amylkar.urrea@udea.edu.co
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.
Wanbo L, Daimin S (1967). “A new compounding life distribution: the Weibull–Poisson distribution.” Journal of Applied Statistics, 9(1), 21–38. doi:10.1080/02664763.2011.575126, https://doi.org/10.1080/02664763.2011.575126.
Examples
old_par <- par(mfrow = c(1, 1)) # save previous graphical parameters
## The probability density function
curve(dWP(x, mu=1.5, sigma=0.5, nu=10), 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(pWP(x, mu=1.5, sigma=0.5, nu=10),
from=0.0001, to=2, col="red", las=1, ylab="F(x)")
curve(pWP(x, mu=1.5, sigma=0.5, nu=10, 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=qWP(p, mu=1.5, sigma=0.5, nu=10), y=p, xlab="Quantile",
las=1, ylab="Probability")
curve(pWP(x, mu=1.5, sigma=0.5, nu=10),
from=0, add=TRUE, col="red")
## The random function
hist(rWP(n=10000, mu=1.5, sigma=0.5, nu=10), freq=FALSE,
xlab="x", ylim=c(0, 2.2), las=1, main="")
curve(dWP(x, mu=1.5, sigma=0.5, nu=10),
from=0.001, to=4, add=TRUE, col="red")
## The Hazard function
curve(hWP(x, mu=1.5, sigma=0.5, nu=10), from=0.001, to=5,
col="red", ylab="Hazard function", las=1)
par(old_par) # restore previous graphical parameters
Electronic equipment data
Description
Time to failure in hours of 18 units of the same electronic device.
Usage
data(equipment)
Format
A vector with 18 observations.
Examples
data(equipment)
hist(equipment, main="", xlab="Time (h)")
initValuesFWE
Description
This function generates initial values for FWE distribution.
Usage
initValuesFWE(y)
Arguments
y |
vector with the random sample |
Value
A two-length numeric vector with initial estimates for mu and sigma
parameters from FWE distribution (see dFWE).
Initial values and search region for Odd Weibull distribution
Description
This function can be used so as to get suggestions about initial values
and the search region for parameter estimation in OW distribution.
Usage
initValuesOW(
formula,
data = NULL,
local_reg = loess.options(),
interpolation = interp.options(),
...
)
Arguments
formula |
an object of class |
data |
an optional data frame containing the response variables. If
data is not specified, the variables are taken from the
environment from which |
local_reg |
a list of control parameters for LOESS. See
|
interpolation |
a list of control parameters for interpolation function. See
|
... |
further arguments passed to
|
Details
This function performs a non-parametric estimation of the empirical total time on test (TTT) plot. Then, this estimated curve can be used so as to get suggestions about initial values and the search region for parameters based on hazard shape associated to the shape of empirical TTT plot.
Value
Returns an object of class c("initValOW", "HazardShape") containing:
-
sigma.startvalue forsigmaparameter of OW distribution. -
nu.startvalue fornuparameter of OW distribution. -
sigma.validsearch region forsigmaparameter of OW distribution. -
nu.validsearch region fornuparameter of OW distribution. -
TTTplotTotal Time on Test transform computed from the data. -
hazard_typeshape of the hazard function determined from the TTT plot.
Author(s)
Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co
Examples
# Example 1
# Bathtuh hazard and its corresponding TTT plot
y1 <- rOW(n = 1000, mu = 0.1, sigma = 7, nu = 0.08)
my_initial_guess1 <- initValuesOW(formula=y1~1)
summary(my_initial_guess1)
plot(my_initial_guess1, par_plot=list(mar=c(3.7,3.7,1,2.5),
mgp=c(2.5,1,0)))
curve(hOW(x, mu = 0.022, sigma = 8, nu = 0.01), from = 0,
to = 80, ylim = c(0, 0.04), col = "red",
ylab = "Hazard function", las = 1)
# Example 2
# Bathtuh hazard and its corresponding TTT plot with right censored data
y2 <- rOW(n = 1000, mu = 0.1, sigma = 7, nu = 0.08)
status <- c(rep(1, 980), rep(0, 20))
my_initial_guess2 <- initValuesOW(formula=Surv(y2, status)~1)
summary(my_initial_guess2)
plot(my_initial_guess2, par_plot=list(mar=c(3.7,3.7,1,2.5),
mgp=c(2.5,1,0)))
curve(hOW(x, mu = 0.022, sigma = 8, nu = 0.01), from = 0,
to = 80, ylim = c(0, 0.04), col = "red",
ylab = "Hazard function", las = 1)
Mice mortality data
Description
The ages at death in weeks for male mice exposed to 240r of gamma radiation.
Usage
data(mice)
Format
A vector with 208 data points.
Examples
data(mice)
hist(mice, main="", xlab="Time (weeks)", freq=FALSE)
lines(density(mice), col="blue", lwd=2)
Custimized region search for odd Weibull distribution
Description
This function can be used to modify OW gamlss.family object
in order to set a customized region search for gamlss() function.
Usage
myOW_region(family = OW, valid.values = "auto", initVal)
Arguments
family |
The |
valid.values |
a list of character elements specifying the region for
|
initVal |
An |
Details
This function was created to help users to fit OW distribution easily
bounding the parametric space for sigma and nu.
The valid.values must be defined as a list of characters containing a call
of the all function.
Value
Returns a gamlss.family object which can be used to fit an OW
distribution in the gamlss() function.
Author(s)
Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co
Examples
# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rOW(n=200, mu=0.2, sigma=4, nu=0.05)
# Custom search region
myvalues <- list(sigma="all(sigma > 1)",
nu="all(nu < 1) & all(nu < 1)")
my_initial_guess <- initValuesOW(formula=y~1)
summary(my_initial_guess)
# OW family modified with 'myOW_region'
require(gamlss)
myOW <- myOW_region(valid.values=myvalues, initVal=my_initial_guess)
mod1 <- gamlss(y~1, sigma.fo=~1, nu.fo=~1,
sigma.start=param.startOW('sigma', my_initial_guess),
nu.start=param.startOW('nu', my_initial_guess),
control=gamlss.control(n.cyc=300, trace=FALSE),
family=myOW)
exp(coef(mod1, what='mu'))
exp(coef(mod1, what='sigma'))
exp(coef(mod1, what='nu'))
# Example 2
# Same example using another link function and using 'myOW_region'
# in the argument 'family'
mod2 <- gamlss(y~1, sigma.fo=~1, nu.fo=~1,
sigma.start=2, nu.start=0.1,
control=gamlss.control(n.cyc=300, trace=FALSE),
family=myOW_region(family=OW(sigma.link='identity'),
valid.values=myvalues,
initVal=my_initial_guess))
exp(coef(mod2, what='mu'))
coef(mod2, what='sigma')
exp(coef(mod2, what='nu'))
Initial values extraction for Odd Weibull distribution
Description
This function can be used to extract initial values found with empirical
time on test transform (TTT) through initValuesOW function.
This is used for parameter estimation in OW distribution.
Usage
param.startOW(param, initValOW)
Arguments
param |
a character used to specify the parameter required. It can take the
values |
initValOW |
an |
Details
This function just gets initial values computed with initValuesOW
for OW family. It must be called in sigma.start and nu.start
arguments from gamlss function. This function is useful only
if user want to set start values automatically with TTT plot.
See example for an illustration.
Value
A length-one vector numeric value corresponding to the initial value of the
parameter specified in param extracted from a initValuesOW
object specified in the initValOW input argument.
Author(s)
Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co
Examples
# Random data generation (OW distributed)
y <- rOW(n=500, mu=0.05, sigma=0.6, nu=2)
# Initial values with TTT plot
iv <- initValuesOW(formula = y ~ 1)
summary(iv)
# This data is from unimodal hazard
# See TTT estimate from sample
plot(iv, legend_options=list(pos=1.03))
# See the true hazard
curve(hOW(x, mu=0.05, sigma=0.6, nu=2), to=100, lwd=3, ylab="h(x)")
# Finally, we fit the model
require(gamlss)
con.out <- gamlss.control(n.cyc = 300, trace = FALSE)
con.in <- glim.control(cyc = 300)
(sigma.start <- param.startOW("sigma", iv))
(nu.start <- param.startOW("nu", iv))
mod <- gamlss(y~1, sigma.fo=~1, nu.fo=~1, control=con.out, i.control=con.in,
family=myOW_region(OW(sigma.link="identity", nu.link="identity"),
valid.values="auto", iv),
sigma.start=sigma.start, nu.start=nu.start)
# Estimates are close to actual values
(mu <- exp(coef(mod, what = "mu")))
(sigma <- coef(mod, what = "sigma"))
(nu <- coef(mod, what = "nu"))
Summary of initValOW objects
Description
This summary method displays initial values and search regions
for OW family.
Usage
## S3 method for class 'initValOW'
summary(object, ...)
Arguments
object |
an object of class |
... |
extra arguments |
Value
No return value, it just prints out in the console the initial values and the
search regions for sigma and nu from OW distribution (see
dOW).
Author(s)
Jaime Mosquera Gutiérrez jmosquerag@unal.edu.co