Title:	Estimation for some Reliability Distributions
Version:	1.0.1
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, 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://fhernanb.github.io/RelDists/
BugReports:	https://github.com/fhernanb/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, alr4
RoxygenNote:	7.3.3
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2026-02-21 16:07:56 UTC; fhern
Author:	Freddy Hernandez-Barajas [aut, cre], Olga Usuga [aut], Carmen Patino [aut], Jaime Mosquera [aut]
Maintainer:	Freddy Hernandez-Barajas <fhernanb@unal.edu.co>
Repository:	CRAN
Date/Publication:	2026-02-24 09:30:26 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

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, S. J. (2018). A reduced new modified Weibull distribution. Communications in Statistics-Theory and Methods, 47(10), 2297-2313.

Xie, M., & Lai, C. D. (1996). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 52(1), 87-93.

See Also

dAddW

Examples

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

# Fitting the model
require(gamlss)

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

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

## End(Not run)

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

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

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

## End(Not run)

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

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, S. J. (2018). A reduced new modified Weibull distribution. Communications in Statistics-Theory and Methods, 47(10), 2297-2313.

Barreto-Souza, W., Santos, A. H., & Cordeiro, G. M. (2010). The beta generalized exponential distribution. Journal of statistical Computation and Simulation, 80(2), 159-172.

See Also

dBGE

Examples

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

# Fitting the model
require(gamlss)

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

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

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

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

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

The Birnbaum-Saunders family

Description

The function BS() defines The Birnbaum-Saunders, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders with parameters mu and sigma has density given by

f(x|\mu,\sigma) = \frac{x^{-3/2}(x+\mu)}{2\sigma\sqrt{2\pi\mu}} \exp\left(\frac{-1}{2\sigma^2}(\frac{x}{\mu}+\frac{\mu}{x}-2)\right)

for x>0, \mu>0 and \sigma>0. In this parameterization \mu is the median of X, E(X)=\mu(1+\sigma^2/2) and Var(X)=(\mu\sigma)^2(1+5\sigma^2/4). The functions proposed here corresponds to the functions created by Roquim et al. (2021) with minor modifications to obtain correct log-likelihoods and random samples.

Value

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

References

Birnbaum, Z.W. and Saunders, S.C. (1969a). A new family of life distributions. J. Appl. Prob., 6, 319–327.

Roquim, F. V., Ramires, T. G., Nakamura, L. R., Righetto, A. J., Lima, R. R., & Gomes, R. A. (2021). Building flexible regression models: including the Birnbaum-Saunders distribution in the gamlss package. Semina: Ciencias Exatas e Tecnologicas, 42(2), 163-168.

See Also

dBS

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rBS(n=100, mu=0.75, sigma=1.3)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS)

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))

# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y as BS
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)
  sigma <- exp(2 - 1.5 * x2)
  y <- rBS(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=300)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS, data=dat)

summary(mod2)

# Example 3
# Fatigue life (T) measures in cycles (×10-3) of n 101
# aluminum coupons (specimens) of type 6061-T6.
# Taken from Leiva et al. (2006) page 37.
# https://journal.r-project.org/articles/RN-2006-033/RN-2006-033.pdf

y <- c(70, 90, 96, 97, 99, 100, 103, 104,
       104, 105, 107, 108, 108, 108, 109, 109,
       112, 112, 113, 114, 114, 114, 116, 119,
       120, 120, 120, 121, 121, 123, 124, 124,
       124, 124, 124, 128, 128, 129, 129, 130,
       130, 130, 131, 131, 131, 131, 131, 132,
       132, 132, 133, 134, 134, 134, 134, 134,
       136, 136, 137, 138, 138, 138, 139, 139,
       141, 141, 142, 142, 142, 142, 142, 142,
       144, 144, 145, 146, 148, 148, 149, 151,
       151, 152, 155, 156, 157, 157, 157, 157,
       158, 159, 162, 163, 163, 164, 166, 166,
       168, 170, 174, 196, 212)

mod3 <- gamlss(y~1, sigma.fo=~1, family=BS)

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

# Example 4
# Aggregate payments by the insurer
# in thousand Skr (Swedish currency).
# Taken from Balakrishnan and Kundu (2019) page 65.
# https://onlinelibrary.wiley.com/doi/abs/10.1002/asmb.2348

y <- c(5014, 5855, 6486, 6540, 6656, 6656, 7212, 7541, 7558, 
       7797, 8546, 9345, 11762, 12478, 13624, 14451,
       14940, 14963, 15092, 16203, 16229, 16730, 18027, 
       18343, 19365, 21782, 24248, 29069, 34267, 38993)

y <- y/10000

mod4 <- gamlss(y~1, sigma.fo=~1, family=BS)

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

The Birnbaum-Saunders family - Santos-Neto et al. (2014)

Description

The function BS2() defines The Birnbaum-Saunders, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS2(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders with parameters mu and sigma has density given by

f(x|\mu,\sigma) = \frac{\exp(\sigma/2)\sqrt{\sigma+1}}{4\sqrt{\pi\mu}x^{3/2}} \left[ x + \frac{\mu\sigma}{\sigma+1} \right] \exp\left( \frac{-\sigma}{4} \left(\frac{x(\sigma+1)}{\mu\sigma}+\frac{\mu\sigma}{x(\sigma+1)} \right) \right)

for x>0, \mu>0 and \sigma>0. In this parameterization E(X)=\mu and Var(X)=\frac{\mu^2(2\sigma+5)}{(\sigma+1)^2}.

Value

Returns a gamlss.family object which can be used to fit a BS2 distribution in the gamlss() function.

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Barros, M. (2014). A reparameterized Birnbaum–Saunders distribution and its moments, estimation and applications. REVSTAT-Statistical Journal, 12(3), 247-272.

See Also

dBS2.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rBS2(n=50, mu=5, sigma=3)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS2)

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))

# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ BS2
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)
  sigma <- exp(2 - 1.5 * x2)
  y <- rBS2(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=100)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS2, data=dat)

summary(mod2)

# Example 3
# Household expenditures for food in the United States (US) expressed
# in thousands of US dollars (M$)
# Santos-Neto et al. (2014) page 266.

y <- c(15.998, 16.652, 21.741, 7.431, 10.481, 13.548, 23.256, 17.976, 
       14.161, 8.825, 14.184, 19.604, 13.728, 21.141, 17.446, 9.629, 
       14.005, 9.160, 18.831, 7.641, 13.882, 9.670, 21.604, 10.866, 
       28.980, 10.882, 18.561, 11.629, 18.067, 14.539, 19.192, 25.918, 
       28.833, 15.869, 14.910, 9.550, 23.066, 14.751)

mod3 <- gamlss(y~1, sigma.fo=~1, family=BS2)

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

# Example 4
# lifetimes of 6061-T6 aluminum coupons expressed in cycles (×10-3)
# at a maximum stress level of 3.1 psi (×104), until the failure to occur.
# Santos-Neto et al. (2014) page 267.

y <- c(70, 90, 96, 97, 99, 100, 103, 104, 104, 105, 107, 108, 108, 108, 109,
       109, 112, 112, 113, 114, 114, 114, 116, 119, 120, 120, 120, 121, 121, 
       123, 124, 124, 124, 124, 124, 128, 128, 129, 129, 130, 130, 130, 131, 
       131, 131, 131, 131, 132, 132, 132, 133, 134, 134, 134, 134, 134, 136, 
       136, 137, 138, 138, 138, 139, 139, 141, 141, 142, 142, 142, 142, 142, 
       142, 144, 144, 145, 146, 148, 148, 149, 151, 151, 152, 155, 156, 157, 
       157, 157, 157, 158, 159, 162, 163, 163, 164, 166, 166, 168, 170, 174, 
       196, 212)

mod4 <- gamlss(y~1, sigma.fo=~1, family=BS2)

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

The Birnbaum-Saunders family - Bourguignon & Gallardo (2022)

Description

The function BS3() defines The Birnbaum-Saunders, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS3(mu.link = "log", sigma.link = "logit")

Arguments

Details

The Birnbaum-Saunders with parameters mu and sigma has density given by

f(x|\mu,\sigma) = \frac{(1-\sigma)y+\mu}{2\sqrt{2\pi\mu\sigma(1-\sigma)}y^{3/2}} \exp{\left[ \frac{-1}{2\sigma} \left( \frac{(1-\sigma)y}{\mu} + \frac{\mu}{(1-\sigma)y} - 2 \right) \right]}

for x>0, \mu>0 and 0<\sigma<1. In this parameterization Mode(X)=\mu and Var(X)=(\mu\sigma)^2(1+5\sigma^2/4).

Value

Returns a gamlss.family object which can be used to fit a BS3 distribution in the gamlss() function.

References

Bourguignon, M., & Gallardo, D. I. (2022). A new look at the Birnbaum–Saunders regression model. Applied Stochastic Models in Business and Industry, 38(6), 935-951.

See Also

dBS3.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rBS3(n=50, mu=2, sigma=0.2)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS3)

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))

# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ BS3
## Not run: 
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)
  inv_logit <- function(x) 1 / (1 + exp(-x))
  sigma <- inv_logit(2 - 1.5 * x2)
  y <- rBS3(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(1234)
dat <- gendat(n=100)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS3, data=dat,
               control=gamlss.control(n.cyc=100))

summary(mod2)

## End(Not run)

# Example 3
# The response variable is the ratio between the average
# rent per acre planted with alfalfa and the corresponding 
# average rent for other agricultural uses. The density of
# dairy cows (X2, number per square mile) is the explanatory variable. 
library(alr4)
data("landrent")

landrent$ratio <- landrent$Y / landrent$X1

with(landrent, plot(x=X2, y=ratio))

mod3 <- gamlss(ratio~X2, sigma.fo=~X2, 
               data=landrent, family=BS3)

summary(mod3)
logLik(mod3)

The two-parameter Chris-Jerry distribution family

Description

The function CJ2() defines The two-parameter Chris-Jerry distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

CJ2(mu.link = "log", sigma.link = "log")

Arguments

Details

The two-parameter Chris-Jerry distribution with parameters mu and sigma has density given by

f(x; \sigma, \mu) = \frac{\mu^2}{\sigma \mu + 2} (\sigma + \mu x^2) e^{-\mu x}; \quad x > 0, \quad \mu > 0, \quad \sigma > 0

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

Value

Returns a gamlss.family object which can be used to fit a CJ2 distribution in the gamlss() function.

Author(s)

Manuel Gutierrez Tangarife, mgutierrezta@unal.edu.co

References

Chinedu, Eberechukwu Q., et al. "New lifetime distribution with applications to single acceptance sampling plan and scenarios of increasing hazard rates" Symmetry 15.10 (2023): 188.

See Also

dCJ2

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rCJ2(n=500, mu=1, sigma=1.5)

# Fitting the model
require(gamlss)

mod1 <- gamlss(y~1, sigma.fo=~1, family=CJ2,
               control=gamlss.control(n.cyc=5000, trace=TRUE))

# Extracting the fitted values for mu, sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))

# Example 2
# Generating random values under some model
gendat <- function(n) {
  x1 <- runif(n, min=0, max=5)
  x2 <- runif(n, min=0, max=5)
  mu <- exp(-0.2 + 1.5 * x1)
  sigma <- exp(1 - 0.7 * x2)
  y <- rCJ2(n=n, mu, sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
datos <- gendat(n=500)

mod2 <- gamlss(y~x1, sigma.fo=~x2, family=CJ2, data=datos,
               control=gamlss.control(n.cyc=5000, trace=TRUE))

summary(mod2)

The Cosine Sine Exponential family

Description

The Cosine Sine Exponential family

Usage

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

Arguments

Details

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

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

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

Value

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

Author(s)

Johan David Marin Benjumea, johand.marin@udea.edu.co

References

Chesneau, C., Bakouch, H. S., & Hussain, T. (2019). A new class of probability distributions via cosine and sine functions with applications. Communications in Statistics-Simulation and Computation, 48(8), 2287-2300.

See Also

dCS2e

Examples

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

# Fitting the model
require(gamlss)

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

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

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

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

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

The Extended Exponential Geometric family

Description

The Extended Exponential Geometric family

Usage

EEG(mu.link = "log", sigma.link = "log")

Arguments

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

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

See Also

dEEG

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Cordeiro, G. M., Ortega, E. M., & Silva, G. O. (2011). The exponentiated generalized gamma distribution with application to lifetime data. Journal of statistical computation and simulation, 81(7), 827-842.

See Also

dEGG

Examples

# Example 1
# Generating some random values with
# known mu, sigma, nu and tau

set.seed(123456)
y <- rEGG(n=100, 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


# A function to simulate a data set with Y ~ EGG
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu    <- exp(-0.8 + -3 * x1)
  sigma <- exp(0.77 - 2 * x2)
  nu    <- 10
  tau   <- 1.5
  y <- rEGG(n=n, mu=mu, sigma=sigma, nu=nu, tau)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(12345)
dat <- gendat(n=200)

mod <- gamlss(y~x1, sigma.fo=~x2, nu.fo=~1, tau.fo=~1, 
              family=EGG, data=dat,
              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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Sarhan, A. M., & Apaloo, J. (2013). Exponentiated modified Weibull extension distribution. Reliability Engineering & System Safety, 112, 137-144.

See Also

dEMWEx

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-Nadarajah-Haghighi family

Description

The Extended Odd Frechet-Nadarjad-Hanhighi family

Usage

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

Arguments

Details

The Extended Odd Frechet-Nadarajah-Haghighi 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)

Helber Santiago Padilla, hspadillar@unal.edu.co

References

Nasiru, S. (2018). Extended Odd Fréchet‐G Family of Distributions Journal of Probability and Statistics, 2018(1), 2931326.

See Also

dEOFNH

Examples

# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
set.seed(123)
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 the model
n <- 100
x1 <- runif(n)
x2 <- runif(n)
mu <- exp(0.5 - 1.2 * x1)
sigma <- 2.1
nu <- 0.8
tau <- 1
y <- rEOFNH(n=n, mu, sigma, nu, tau)

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

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

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

dEW

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 exponentiated XLindley family

Description

The function EXL() defines The exponentiated XLindley, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

EXL(mu.link = "log", sigma.link = "log")

Arguments

Details

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

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

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

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

Value

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

Author(s)

Manuel Gutierrez Tangarife, mgutierrezta@unal.edu.co

References

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

See Also

EXL.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rEXL(n=300, mu=0.75, sigma=1.3)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=EXL,
               control=gamlss.control(n.cyc=5000, trace=FALSE))

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))


# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ EXL
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)
  sigma <- exp(2 - 1.5 * x2)
  y <- rEXL(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(1234)
dat <- gendat(n=100)

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

summary(mod2)

# Example 3
# Mortality rate due to COVID-19 for 30 days (31st March to April 30, 2020)
# recorded for the Netherlands.
# Taken from Alomair et al. (2024) page 12.

x <- c(14.918, 10.656, 12.274, 10.289, 10.832, 7.099, 5.928, 13.211, 
       7.968, 7.584, 5.555, 6.027, 4.097, 3.611, 4.960, 7.498, 6.940, 
       5.307, 5.048, 2.857, 2.254, 5.431, 4.462, 3.883,
       3.461, 3.647, 1.974, 1.273, 1.416, 4.235)

mod3 <- gamlss(x~1, sigma.fo=~1, family=EXL,
               control=gamlss.control(n.cyc=5000, trace=FALSE))

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

# Replicating figure 4 from Alomair et al. (2024)
# Hist and estimated pdf
hist(x, freq=FALSE)
curve(dEXL(x, mu=0.4089915, sigma=2.710467), add=TRUE, 
      col="tomato", lwd=2)
# Empirical cdf and estimated ecdf
plot(ecdf(x))
curve(pEXL(x, mu=0.4089915, sigma=2.710467), add=TRUE, 
      col="tomato", lwd=2)
# QQplot
qqplot(x, rEXL(n=30, mu=0.4089915, sigma=2.710467), 
       col="tomato")
qqline(x, distribution=function(p) qEXL(p, mu=0.4089915, sigma=2.710467))


# Example 4
# Precipitation in inches
# Taken from Alomair et al. (2024) page 13.

# Manuel

The Extended Weibull family

Description

The Extended Weibull family

Usage

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

Arguments

Details

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

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

for x > 0.

Value

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Zhang, T., & Xie, M. (2007). Failure data analysis with extended Weibull distribution. Communications in Statistics—Simulation and Computation, 36(3), 579-592.

See Also

dExW

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 Ex-Wald family

Description

The function ExWALD() defines the Ex-wALD distribution, three-parameter continuous distribution for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

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

Arguments

Details

The Ex-Wald distribution with parameters \mu, \sigma and \nu has density given by

f(x |\mu, \sigma, \nu) = \frac{1}{\nu} \exp(\frac{-x}{\nu} + \sigma(\mu-k)) F_W(x|k, \sigma) \, \text{for} \, k \geq 0

f(x |\mu, \sigma, \nu) = \frac{1}{\nu} \exp\left( \frac{-(\sigma-\mu)^2}{2x} \right) Re \left( w(k^\prime \sqrt{x/2} + \frac{\sigma i}{\sqrt{2x}}) \right) \, \text{for} \, k < 0

where k=\sqrt{\mu^2-\frac{2}{\nu}}, k^\prime=\sqrt{\frac{2}{\nu}-\mu^2} and F_W corresponds to the cumulative function of the Wald distribution.

More details about those expressions can be found on page 680 from Heathcote (2004).

Value

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

Author(s)

Freddy Hernandez, fhernanb@unal.edu.co

References

Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times. Behavior Research Methods, Instruments, & Computers, 33, 457-469.

Heathcote, A. (2004). Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package. Behavior Research Methods, Instruments, & Computers, 36, 678-694.

See Also

dExWALD.

Examples

# Example 1
# Generating random values with
# known mu, sigma and nu

mu <- 0.20
sigma <- 70
nu <- 115

set.seed(123)
y <- rExWALD(n=100, mu, sigma, nu)

library(gamlss)
mod1 <- gamlss(y~1, family=ExWALD,
               control=gamlss.control(n.cyc=1000, trace=TRUE))

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

# Example 2
# Generating random values under some model


# A function to simulate a data set with Y ~ ExWALD
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu    <- exp(-1 + 2.8 * x1) # 1.5 approximately
  sigma <- exp( 1 - 1.2 * x2) # 1.5 approximately
  nu    <- 2
  y <- rExWALD(n=n, mu=mu, sigma=sigma, nu=nu)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(1234)
dat <- gendat(n=200)

# Fitting the model
mod2 <- gamlss(y ~ x1,
               sigma.fo = ~ x2,
               nu.fo = ~ 1,
               family = ExWALD,
               data = dat,
               control = gamlss.control(n.cyc=1000, 
                                        trace=TRUE))
summary(mod2)

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

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

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, A. N. (2013). The generalized Gompertz distribution. Applied mathematical modelling, 37(1-2), 13-24.

See Also

dGGD

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

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

Almalki, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

De Gusmao, F. R., Ortega, E. M., & Cordeiro, G. M. (2011). The generalized inverse Weibull distribution. Statistical Papers, 52, 591-619.

See Also

dGIW

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

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

dGMW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Stacy, E. W. (1962). A generalization of the gamma distribution. The Annals of mathematical statistics, 1187-1192.

See Also

dGammaW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Drapella, A. (1993). The complementary Weibull distribution: unknown or just forgotten?. Quality and reliability engineering international, 9(4), 383-385.

See Also

dIW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Shahbaz, M. Q., Shahbaz, S., & Butt, N. S. (2012). The Kumaraswamy Inverse Weibull Distribution. Pakistan journal of statistics and operation research, 479-489.

See Also

dKumIW

Examples

# Example 1
# Generating some random values with
# known mu, sigma, nu and tau
y <- rKumIW(n=100, 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

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, D. V. (1958). Fiducial distributions and Bayes' theorem. Journal of the Royal Statistical Society. Series B (Methodological), 102-107.

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Gumbel, E. J. (1958). Statistics of extremes. Columbia university press.

See Also

dLW

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

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

Okasha, H. M., El-Baz, A. H., Tarabia, A. M. K., & Basheer, A. M. (2017). Extended inverse Weibull distribution with reliability application. Journal of the Egyptian Mathematical Society, 25(3), 343-349.

See Also

dMOEIW

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
set.seed(123456)
y <- rMOEIW(n=100, 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

# A function to simulate a data set with Y ~ MOEIW
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu    <- exp(-2.02 + 3 * x1) # 0.60 approximately
  sigma <- exp(2.23 - 2 * x2)  # 3.42 approximately
  nu    <- 2
  y <- rMOEIW(n=n, mu=mu, sigma=sigma, nu=nu)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=100)

mod <- gamlss(y~x1, sigma.fo=~x2, nu.fo=~1, 
              family=MOEIW, data=dat,
              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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Ghitany, M. E., Al-Hussaini, E. K., & Al-Jarallah, R. A. (2005). Marshall–Olkin extended Weibull distribution and its application to censored data. Journal of Applied Statistics, 32(10), 1025-1034.

See Also

dMOEW

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

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. (2019). The Marshall-Olkin kappa distribution: properties and applications. Journal of King Saud University-Science, 31(4), 684-691.

See Also

dMOK

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Lai, C. D., Xie, M., & Murthy, D. N. P. (2003). A modified Weibull distribution. IEEE Transactions on reliability, 52(1), 33-37.

See Also

dMW

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')

New Exponentiated Exponential family

Description

The function NEE() defines the New Exponentiated Exponential distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

NEE(mu.link = "log", sigma.link = "log")

Arguments

Details

The New Exponentiated Exponential distribution with parameters mu and sigma has density given by

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

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

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

Value

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

References

Hassan, Anwar, I. H. Dar, and M. A. Lone. "A New Class of Probability Distributions With An Application to Engineering Data." Pakistan Journal of Statistics and Operation Research 20.2 (2024): 217-231.

See Also

dNEE

Examples

# Example 1
# Generating some random values with
# known mu and sigma
y <- rNEE(n=500, mu=2.5, sigma=3.5)

# Fitting the model
require(gamlss)

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

# Extracting the fitted values for mu, sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
exp(coef(mod1, what="sigma"))

# Example 2
# Generating random values under some model
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(-0.2 + 1.5 * x1)
  sigma <- exp(1 - 0.7 * x2)
  y <- rNEE(n=n, mu, sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
datos <- gendat(n=500)

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

summary(mod2)

# Example 3  --------------------------------------------------
# Obtained from Hassan (2024) page 226
# The data set consists of 63 observations of the gauge lengths of 10mm. 

y <- c(1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 
       2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532,  2.575, 2.614, 
       2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 
       2.928, 2.937, 2.937, 2.977, 2.996,  3.030, 3.125, 3.139, 3.145, 
       3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 
       3.377, 3.408, 3.435,  3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 
       3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020)

mod3 <- gamlss(y~1, family=NEE)

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

# Hist and estimated pdf
hist(y, freq=FALSE, ylim=c(0, 0.7))
curve(dNEE(x, mu=2.076862, sigma=255.2289), 
      add=TRUE, col="tomato", lwd=2)

# Empirical cdf and estimated ecdf
plot(ecdf(y))
curve(pNEE(x, mu=2.076862, sigma=255.2289), 
      add=TRUE, col="tomato", lwd=2)
# QQplot
qqplot(y, rNEE(n=length(y), mu=2.076862, sigma=255.2289), col="tomato")
qqline(y, distribution=function(p) qNEE(p, mu=2.076862, sigma=255.2289))

# Example 4  --------------------------------------------------
# Obtained from Hassan (2024) page 226
# The dataset was reported by Bader and Priest (1982) on failure 
# stresses (in GPa) of 65 single carbon fibers of lengths 50 mm

y <- c(0.564, 0.729, 0.802, 0.95, 1.053, 1.111, 1.115, 1.194, 1.208,
       1.216, 1.247, 1.256, 1.271, 1.277, 1.305, 1.313, 1.348, 
       1.39, 1.429, 1.474, 1.49, 1.503, 1.52, 1.522, 1.524, 1.551, 
       1.551, 1.609, 1.632, 1.632, 1.676, 1.684, 1.685, 1.728, 1.74, 
       1.761, 1.764, 1.785, 1.804, 1.816, 1.824, 1.836, 1.879, 1.883, 
       1.892, 1.898, 1.934, 1.947, 1.976, 2.02, 2.023, 2.05, 2.059, 
       2.068, 2.071, 2.098, 2.13, 2.204, 2.317, 2.334, 2.34, 2.346, 
       2.378, 2.483, 2.269)

mod4 <- gamlss(y~1, family=NEE)

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

hist(y, freq=FALSE)
curve(dNEE(x, mu=2.400515, sigma=25.15236), 
      add=TRUE, col="tomato", lwd=2)

# Empirical cdf and estimated ecdf
plot(ecdf(y))
curve(pNEE(x, mu=2.400515, sigma=25.15236), 
      add=TRUE, col="tomato", lwd=2)
# QQplot
qqplot(y, rNEE(n=length(y), mu=2.400515, sigma=25.15236), col="tomato")
qqline(y, distribution=function(p) qNEE(p, mu=2.400515, sigma=25.15236))

# Example 5 -------------------------------------------------------------------
# 69 Observations of the gauge lengths of 20m.
y <- c(1.312,1.314,1.479,1.552,1.700,1.803,1.861,1.865,1.944,1.958,1.966,1.997,
       2.006,2.021,2.027,2.055, 2.063,2.098,2.140,2.179,2.224,2.240,2.253,2.270,
       2.272,2.274,2.301,2.301,2.359,2.382,2.382,2.426, 2.434,2.435,2.478,2.490,
       2.511,2.514,2.535,2.554,2.566,2.570,2.586,2.629,2.633,2.642,2.648,2.684,
       2.697,2.726,2.770,2.773,2.800,2.809,2.818,2.821,2.848,2.880,2.954,3.012,
       3.067,3.084,3.090,3.096, 3.128,3.233,3.433,3.585,3.585)

mod5 <- gamlss(y~1, sigma.fo=~1, family = NEE)

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

hist(y, freq=FALSE)
curve(dNEE(x, mu=2.197771, sigma=100.8888), add=TRUE, 
      col="tomato", lwd=2)
# Empirical cdf and estimated ecdf
plot(ecdf(y))
curve(pNEE(x, mu=2.197771, sigma=100.8888), add=TRUE, 
      col="tomato", lwd=2)
# QQplot
qqplot(y, rNEE(n=length(y), mu=2.197771, sigma=100.8888), col="tomato")
qqline(y, distribution=function(p) qNEE(p, mu=2.197771, sigma=100.8888))

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

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.

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 <- 200
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)
y <- rOW(n=n, mu, sigma, nu)

mod <- gamlss(y~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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Ghitany, M. E., Al-Mutairi, D. K., Balakrishnan, N., & Al-Enezi, L. J. (2013). Power Lindley distribution and associated inference. Computational Statistics & Data Analysis, 64, 20-33.

See Also

dPL

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

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

Sen, S., Korkmaz, M. Ç., & Yousof, H. M. (2018). The quasi XGamma-Poisson distribution: properties and application. Istatistik Journal of The Turkish Statistical Association, 11(3), 65-76.

See Also

dQXGP

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Cohen, A. C. (1973). The reflected Weibull distribution. Technometrics, 15(4), 867-873.

See Also

dRW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Sarhan, A. M., & Zaindin, M. (2009). Modified Weibull distribution. APPS. Applied Sciences, 11, 123-136.

See Also

dSZMW

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 Wald family

Description

The function WALD() defines the wALD distribution, two-parameter continuous distribution for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

WALD(mu.link = "log", sigma.link = "log")

Arguments

Details

The Wald distribution with parameters \mu and sigma has density given by

\operatorname{f}(x |\mu, \sigma)=\frac{\sigma}{\sqrt{2 \pi x^3}} \exp \left[-\frac{(\sigma-\mu x)^2}{2x}\right ], x>0

Value

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

Author(s)

Sofia Cuartas García, scuartasg@unal.edu.co

References

Heathcote, A. (2004). Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package. Behavior Research Methods, Instruments, & Computers, 36, 678-694.

See Also

dWALD.

Examples

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

y <- rWALD(10000, mu, sigma)

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

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

# Example 2
# Generating random values under some model

# A function to simulate a data set with Y ~ WALD
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(0.75 - 0.69 * x1)   # Approx 1.5
  sigma <- exp(0.5 - 0.64 * x2) # Approx 1.20
  y <- rWALD(n, mu, sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

dat <- gendat(n=200)

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

summary(mod2)

The Weibull Geometric family

Description

The Weibull Geometric distribution

Usage

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

Arguments

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, A. L., & Cordeiro, G. M. (2011). The Weibull-geometric distribution. Journal of Statistical Computation and Simulation, 81(5), 645-657.

See Also

dWG

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

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

dWGEE

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

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

Lu, Wanbo, and Daimin Shi. "A new compounding life distribution: the Weibull–Poisson distribution." Journal of applied statistics 39.1 (2012): 21-38.

See Also

dWP

Examples

# Example 1
# Generating some random values with
# known mu, sigma and nu
y <- rWP(n=3000, mu=1.5, sigma=0.5, nu=0.5)

# Fitting the model
require(gamlss)

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

# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ WP
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(-1.3 + 3.1 * x1)
  sigma <- exp(0.9 - 3.2 * x2)
  nu <- 0.5
  y <- rWP(n=n, mu, sigma, nu)
  data.frame(y=y, x1=x1, x2)
}

set.seed(1234)
dat <- gendat(n=100)

# Fitting the model
mod2 <- NULL
mod2 <- gamlss(y~x1, sigma.fo=~x2, nu.fo=~1, 
               family=WP, data=dat,
              control=gamlss.control(n.cyc=5000, trace=FALSE))

coef(mod2, what="mu")
coef(mod2, what="sigma")
exp(coef(mod2, 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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Xie, M., & Lai, C. D. (1996). Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 52(1), 87-93.

See Also

AddW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Barreto-Souza, W., Santos, A. H., & Cordeiro, G. M. (2010). The beta generalized exponential distribution. Journal of statistical Computation and Simulation, 80(2), 159-172.

See Also

BGE

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 Birnbaum-Saunders distribution

Description

Density, distribution function, quantile function, random generation and hazard function for the Birnbaum-Saunders distribution with parameters mu and sigma.

Usage

dBS(x, mu = 1, sigma = 1, log = FALSE)

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

qBS(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rBS(n, mu = 1, sigma = 1)

hBS(x, mu, sigma)

Arguments

Details

The Birnbaum-Saunders with parameters mu and sigma has density given by

f(x|\mu,\sigma) = \frac{x^{-3/2}(x+\mu)}{2\sigma\sqrt{2\pi\mu}} \exp\left(\frac{-1}{2\sigma^2}(\frac{x}{\mu}+\frac{\mu}{x}-2)\right)

for x>0, \mu>0 and \sigma>0. In this parameterization \mu is the median of X, E(X)=\mu(1+\sigma^2/2) and Var(X)=(\mu\sigma)^2(1+5\sigma^2/4). The functions proposed here corresponds to the functions created by Roquim et al. (2021) with minor modifications to obtain correct log-likelihoods and random samples.

Value

dBS gives the density, pBS gives the distribution function, qBS gives the quantile function, rBS generates random deviates and hBS gives the hazard function.

References

Birnbaum, Z.W. and Saunders, S.C. (1969a). A new family of life distributions. J. Appl. Prob., 6, 319–327.

Roquim, F. V., Ramires, T. G., Nakamura, L. R., Righetto, A. J., Lima, R. R., & Gomes, R. A. (2021). Building flexible regression models: including the Birnbaum-Saunders distribution in the gamlss package. Semina: Ciencias Exatas e Tecnologicas, 42(2), 163-168.

See Also

BS.

Examples

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


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


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

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

#example 3
## The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qBS(p, mu=2.3, sigma=1.7), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS(x, mu=2.3, sigma=1.7), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

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

#example 4
## The random function
x <- rBS(n=10000, mu=20, sigma=0.5)
hist(x, freq=FALSE)
curve(dBS(x, mu=20, sigma=0.5), from=0, to=100, 
      add=TRUE, col="tomato", lwd=2)

#example 5
## The Hazard function
curve(hBS(x, mu=20, sigma=0.5), from=0.001, to=100,
      col="tomato", ylab="Hazard function", las=1)

The Birnbaum-Saunders distribution - Santos-Neto et al. (2014)

Description

Density, distribution function, quantile function, random generation and hazard function for the Birnbaum-Saunders distribution with parameters mu and sigma.

Usage

dBS2(x, mu = 1, sigma = 1, log = FALSE)

pBS2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qBS2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rBS2(n, mu = 1, sigma = 1)

hBS2(x, mu, sigma)

Arguments

Details

The Birnbaum-Saunders with parameters mu and sigma has density given by

f(x) = \frac{\exp(\sigma/2)\sqrt{\sigma+1}}{4\sqrt{\pi\mu}x^{3/2}} \left[ x + \frac{\mu\sigma}{\sigma+1} \right] \exp\left( \frac{-\sigma}{4} \left(\frac{x(\sigma+1)}{\mu\sigma}+\frac{\mu\sigma}{x(\sigma+1)} \right) \right)

for x>0, \mu>0 and \sigma>0. In this parameterization E(X)=\mu and Var(X)=(\mu\sigma)^2(1+5\sigma^2/4). The functions proposed here corresponds to the parameterization proposed by Santos-Neto et al. (2014).

Value

dBS2 gives the density, pBS2 gives the distribution function, qBS2 gives the quantile function, rBS2 generates random deviates and hBS2 gives the hazard function.

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Barros, M. (2014). A reparameterized Birnbaum–Saunders distribution and its moments, estimation and applications. REVSTAT-Statistical Journal, 12(3), 247-272.

See Also

BS2.

Examples

#Example 1
#Plotting the mass function for different parameter values
curve(dBS2(x, mu=1.0, sigma=100), 
      from=0.001, to=5,
      ylim=c(0, 3), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS2(x, mu=1.5, sigma=100),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dBS2(x, mu=2.0, sigma=100),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=1.0, sigma=100", 
                            "mu=1.5, sigma=100",
                            "mu=2.0, sigma=100"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


curve(dBS2(x, mu=1, sigma=2), 
      from=0.001, to=2,
      ylim=c(0, 1.1), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS2(x, mu=1, sigma=5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(dBS2(x, mu=1, sigma=10),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=1, sigma=2", 
                            "mu=1, sigma=5",
                            "mu=1, sigma=10"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.6)


# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS2(x, mu=0.5, sigma=0.5), 
      from=0.001, to=15,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="f(x)")
curve(pBS2(x, mu=1, sigma=0.5),
      col="tomato", 
      lwd=2,
      add=TRUE)
curve(pBS2(x, mu=1.5, sigma=0.5),
      col="seagreen",
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=0.5, sigma=0.5", 
                               "mu=1.0, sigma=0.5",
                               "mu=1.5, sigma=0.5"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.5)

# Example 3
# The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qBS2(p, mu=2.3, sigma=1.7), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS2(x, mu=2.3, sigma=1.7), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

# Example 4
# The random function
x <- rBS2(n=10000, mu=2.5, sigma=100)
hist(x, freq=FALSE)
curve(dBS2(x, mu=2.5, sigma=100), from=0, to=10, 
      add=TRUE, col="tomato", lwd=2)

# Example 5
# The Hazard function
curve(hBS2(x, mu=20, sigma=0.5), from=0.001, to=100,
      col="tomato", ylab="Hazard function", las=1)

The Birnbaum-Saunders distribution - Bourguignon & Gallardo (2022)

Description

Density, distribution function, quantile function, random generation and hazard function for the Birnbaum-Saunders distribution with parameters mu and sigma.

Usage

dBS3(x, mu = 1, sigma = 0.5, log = FALSE)

pBS3(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

qBS3(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

rBS3(n, mu = 1, sigma = 0.5)

hBS3(x, mu, sigma)

Arguments

Details

The Birnbaum-Saunders with parameters mu and sigma has density given by

f(x) = \frac{\exp(\sigma/2)\sqrt{\sigma+1}}{4\sqrt{\pi\mu}x^{3/2}} \left[ x + \frac{\mu\sigma}{\sigma+1} \right] \exp\left( \frac{-\sigma}{4} \left(\frac{x(\sigma+1)}{\mu\sigma}+\frac{\mu\sigma}{x(\sigma+1)} \right) \right)

for x>0, \mu>0 and \sigma>0. In this parameterization E(X)=\mu and Var(X)=(\mu\sigma)^2(1+5\sigma^2/4). The functions proposed here corresponds to the parameterization proposed by Santos-Neto et al. (2014).

Value

dBS3 gives the density, pBS3 gives the distribution function, qBS3 gives the quantile function, rBS3 generates random deviates and hBS3 gives the hazard function.

References

Bourguignon, M., & Gallardo, D. I. (2022). A new look at the Birnbaum–Saunders regression model. Applied Stochastic Models in Business and Industry, 38(6), 935-951.

See Also

BS3.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dBS3(x, mu=2, sigma=0.2), 
      from=0.001, to=10,
      ylim=c(0, 0.4), 
      col="royalblue1", lwd=2, 
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dBS3(x, mu=2, sigma=0.4),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=2, sigma=0.2", 
                            "mu=2, sigma=0.4"),
       col=c("royalblue1", "tomato"), lwd=2, cex=0.6)

# Example 2
# Checking if the cumulative curves converge to 1
curve(pBS3(x, mu=2, sigma=0.2), 
      from=0.00001, to=10,
      ylim=c(0, 1), 
      col="royalblue1", lwd=2, 
      main="Cumulative Distribution Function",
      xlab="x", ylab="f(x)")
curve(pBS3(x, mu=2, sigma=0.4),
      col="tomato", 
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=2, sigma=0.2", 
                               "mu=2, sigma=0.4"),
       col=c("royalblue1", "tomato", "seagreen"), lwd=2, cex=0.5)

# Example 3
# The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qBS3(p, mu=2, sigma=0.2), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pBS3(x, mu=2, sigma=0.2), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

# Example 4
# The random function
x <- rBS3(n=10000, mu=2, sigma=0.2)
hist(x, freq=FALSE)
curve(dBS3(x, mu=2, sigma=0.2), from=0, to=10, 
      add=TRUE, col="tomato", lwd=2)

# Example 5
# The Hazard function
curve(hBS3(x, mu=2, sigma=0.2), from=0.001, to=4,
      col="tomato", ylab="Hazard function", las=1)

The two-parameter Chris-Jerry distribution

Description

Density, distribution function, quantile function, random generation and hazard function for the two-parameter Chris-Jerry distribution with parameters mu and sigma.

Usage

dCJ2(x, mu, sigma, log = FALSE)

pCJ2(q, mu, sigma, log.p = FALSE, lower.tail = TRUE)

qCJ2(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)

rCJ2(n, mu, sigma)

hCJ2(x, mu, sigma, log = FALSE)

Arguments

Details

The two-parameter Chris-Jerry distribution with parameters mu and sigma has density given by

f(x; \sigma, \mu) = \frac{\mu^2}{\sigma \mu + 2} (\sigma + \mu x^2) e^{-\mu x}; \quad x > 0, \quad \mu > 0, \quad \sigma > 0

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

Value

dCJ2 gives the density, pCJ2 gives the distribution function, qCJ2 gives the quantile function, rCJ2 generates random deviates and hCJ2 gives the hazard function.

Author(s)

Manuel Gutierrez Tangarife, mgutierrezta@unal.edu.co

References

Chinedu, Eberechukwu Q., et al. "New lifetime distribution with applications to single acceptance sampling plan and scenarios of increasing hazard rates" Symmetry 15.10 (2023): 188.

See Also

CJ2

Examples

# Example 1
# Plotting the density function for different parameter values
curve(dCJ2(x, mu=3.5, sigma=0.01), 
      from=0.0001, to=5,
      ylim=c(0, 1),
      col="red", lwd=2,
      main="Density function",
      xlab="x", ylab="f(x)")
curve(dCJ2(x, mu=2, sigma=0.05),
      col="green",
      lwd=2,
      add=TRUE)
curve(dCJ2(x, mu=1.5, sigma=0.01),
      col="blue",
      lwd=2,
      add=TRUE)
curve(dCJ2(x, mu=2.5, sigma=0.01),
      col="lightblue",
      lwd=2,
      add=TRUE)
legend("topright", legend=c("mu=3.5, sigma=0.01",
                            "mu=2, sigma=0.05",
                            "mu=1.5, sigma=0.01",
                            "mu=2.5, sigma=0.1"),
       col=c( "red", "green","blue","lightblue"), lwd=2, cex=0.6)

# Example 2
# Checking if the cumulative curves converge to 1
curve(pCJ2(x, mu=2.7, sigma=0.1),
      from=0.0001, to=5,
      ylim=c(0, 1),
      col="red", lwd=2,
      main="Cumulative function",
      xlab="x", ylab="f(x)")
curve(pCJ2(x, mu=2.3, sigma=0.5),
      col="green",
      lwd=2,
      add=TRUE)
curve(pCJ2(x, mu=2.8, sigma=0.2),
      col="blue",
      lwd=2,
      add=TRUE)
curve(pCJ2(x, mu=3.8, sigma=0.3),
      col="lightblue",
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=2.75, sigma=0.1",
                            "mu=2.3, sigma=0.5",
                            "mu=2.8, sigma=0.2",
                            "mu=3.8, sigma=0.3"),
       col=c( "red", "green","blue","lightblue"), lwd=2, cex=0.6)

# Example 3
# Checking the quantile function
p <- seq(from=0.0001, to=0.99999, length.out=100)
plot(x=qCJ2(p, mu=2.3, sigma=1.7), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pCJ2(x, mu=2.3, sigma=1.7), 
      from=0.0001, add=TRUE, col="red", lwd=2.5)

# Example 4
# Comparing the random generator output with
# the theoretical probabilities
x <- rCJ2(n=10000, mu=1.5, sigma=2.5)
hist(x, freq=FALSE)
curve(dCJ2(x, mu=1.5, sigma=2.5), from=0.001, to=8, 
      add=TRUE, col="tomato", lwd=2)

# Example 5
# The Hazard function
curve(hCJ2(x, mu=0.85, sigma=0.15),
      from=0.0001, to=5,
      ylim=c(0, 1),
      col="red", lwd=2,
      main="Hazard function",
      xlab="x", ylab="f(x)")
curve(hCJ2(x, mu=1, sigma=0.05),
      col="green",
      lwd=2,
      add=TRUE)
curve(hCJ2(x, mu=0.9, sigma=0.1),
      col="blue",
      lwd=2,
      add=TRUE)
curve(hCJ2(x, mu=1.15, sigma=0.1),
      col="lightblue",
      lwd=2,
      add=TRUE)
legend("bottomright", legend=c("mu=0.85, sigma=0.15",
                               "mu=1, sigma=0.05",
                               "mu=0.9, sigma=0.1",
                               "mu=1.15, sigma=0.1"),
       col=c( "red", "green","blue","lightblue"), lwd=2, cex=0.5)

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

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, H. S., & Hussain, T. (2019). A new class of probability distributions via cosine and sine functions with applications. Communications in Statistics-Simulation and Computation, 48(8), 2287-2300.

See Also

CS2e

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

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

See Also

EEG

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Cordeiro, G. M., Ortega, E. M., & Silva, G. O. (2011). The exponentiated generalized gamma distribution with application to lifetime data. Journal of statistical computation and simulation, 81(7), 827-842.

See Also

EGG

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 Modified 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

Details

The Exponentiated Modified 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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Sarhan, A. M., & Apaloo, J. (2013). Exponentiated modified Weibull extension distribution. Reliability Engineering & System Safety, 112, 137-144.

See Also

EMWEx

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

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, hspadillar@unal.edu.co

References

Nasiru, S. (2018). Extended Odd Fréchet‐G Family of Distributions Journal of Probability and Statistics, 2018(1), 2931326.

See Also

EOFNH

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

Details

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

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

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

Value

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

See Also

EW

Examples

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

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

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

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

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

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

par(old_par) # restore previous graphical parameters

The exponentiated XLindley distribution

Description

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

Usage

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

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

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

rEXL(n, mu, sigma)

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

Arguments

Details

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

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

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

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

Value

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

Author(s)

Manuel Gutierrez Tangarife, mgutierrezta@unal.edu.co

References

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

See Also

EXL.

Examples

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


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


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

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


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

# Comparing quantile, density and cumulative
p <- c(0.25, 0.5, 0.75)
quantile <- qEXL(p=p, mu=2.3, sigma=1.7) 

for(i in quantile){
  print(integrate(dEXL, lower=0, upper=i, mu=2.3, sigma=1.7))
}

pEXL(q=quantile, mu=2.3, sigma=1.7)

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

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

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Zhang, T., & Xie, M. (2007). Failure data analysis with extended Weibull distribution. Communications in Statistics—Simulation and Computation, 36(3), 579-592.

See Also

ExW

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 Ex-Wald distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Ex-Wald distribution with parameter \mu, \sigma and \nu.

Usage

dExWALD(x, mu = 1.5, sigma = 1.5, nu = 2, log = FALSE)

pExWALD(q, mu = 1.5, sigma = 1.5, nu = 2, lower.tail = TRUE, log.p = FALSE)

qExWALD(p, mu = 1.5, sigma = 1.5, nu = 2)

rExWALD(n, mu = 1.5, sigma = 1.5, nu = 2)

Arguments

Details

The Wald distribution with parameters \mu, \sigma and \nu has density given by

f(x |\mu, \sigma, \nu) = \frac{1}{\nu} \exp(\frac{-x}{\nu} + \sigma(\mu-k)) F_W(x|k, \sigma) \, \text{for} \, k \geq 0

f(x |\mu, \sigma, \nu) = \frac{1}{\nu} \exp\left( \frac{-(\sigma-\mu)^2}{2x} \right) Re \left( w(k^\prime \sqrt{x/2} + \frac{\sigma i}{\sqrt{2x}}) \right) \, \text{for} \, k < 0

where k=\sqrt{\mu^2-\frac{2}{\nu}}, k^\prime=\sqrt{\frac{2}{\nu}-\mu^2} and F_W corresponds to the cumulative function of the Wald distribution.

More details about those expressions can be found on page 680 from Heathcote (2004).

Value

dExWALD gives the density, pExWALD gives the distribution function, qExWALD gives the quantile function, rExWALD generates random deviates.

Author(s)

Freddy Hernandez, fhernanb@unal.edu.co

References

Schwarz, W. (2001). The ex-Wald distribution as a descriptive model of response times. Behavior Research Methods, Instruments, & Computers, 33, 457-469.

Heathcote, A. (2004). Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package. Behavior Research Methods, Instruments, & Computers, 36, 678-694.

See Also

ExWALD

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dExWALD(x, mu=0.15, sigma=52.5, nu=50), ylim=c(0, 0.005),
      from=0, to=1200, col="cadetblue3", las=1, ylab="f(x)")

curve(dExWALD(x, mu=0.20, sigma=70, nu=50),
      add=TRUE, col= "purple")

curve(dExWALD(x, mu=0.25, sigma=87.5, nu=50),
      add=TRUE, col="goldenrod")

curve(dExWALD(x, mu=0.20, sigma=70, nu=115),
      add=TRUE, col="tomato")

curve(dExWALD(x, mu=0.20, sigma=70, nu=35),
      add=TRUE, col="blue")

legend("topright", col=c("cadetblue3", "purple", "goldenrod",
                         "tomato", "blue"), 
       lty=1, bty="n",
       legend=c("mu=0.15, sigma=52.5, nu=50",
                "mu=0.20, sigma=70.0, nu=50",
                "mu=0.25, sigma=87.5, nu=50",
                "mu=0.20, sigma=70.0, nu=115",
                "mu=0.20, sigma=70.0, nu=35"))

# Example 2
# Checking if the cumulative curves converge to 1
curve(pExWALD(x, mu=0.15, sigma=52.5, nu=50), ylim=c(0, 1),
      from=0, to=1200, col="cadetblue3", las=1, ylab="F(x)")

curve(pExWALD(x, mu=0.20, sigma=70, nu=50),
      add=TRUE, col= "purple")

curve(pExWALD(x, mu=0.25, sigma=87.5, nu=50),
      add=TRUE, col="goldenrod")

curve(pExWALD(x, mu=0.20, sigma=70, nu=115),
      add=TRUE, col="tomato")

curve(pExWALD(x, mu=0.20, sigma=70, nu=35),
      add=TRUE, col="blue")

legend("bottomright", col=c("cadetblue3", "purple", "goldenrod",
                         "tomato", "blue"), 
       lty=1, bty="n",
       legend=c("mu=0.15, sigma=52.5, nu=50",
                "mu=0.20, sigma=70.0, nu=50",
                "mu=0.25, sigma=87.5, nu=50",
                "mu=0.20, sigma=70.0, nu=115",
                "mu=0.20, sigma=70.0, nu=35"))

# Example 3
# Checking the quantile function
mu <- 5
sigma <- 3
nu <- 2
p <- seq(from=0.1, to=0.99, length.out=100)
plot(x=qExWALD(p, mu=mu, sigma=sigma, nu=nu), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pExWALD(x, mu=mu, sigma=sigma, nu=nu), from=0, add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical probabilities
mu <- 0.2
sigma <- 70
nu <- 35
x <- rExWALD(n=10000, mu=mu, sigma=sigma, nu=nu)
hist(x, freq=FALSE)
curve(dExWALD(x, mu=mu, sigma=sigma, nu=nu), col="tomato", add=TRUE)

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

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.

See Also

FWE

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

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, A. N. (2013). The generalized Gompertz distribution. Applied mathematical modelling, 37(1-2), 13-24.

See Also

GGD

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

De Gusmao, F. R., Ortega, E. M., & Cordeiro, G. M. (2011). The generalized inverse Weibull distribution. Statistical Papers, 52, 591-619.

See Also

GIW

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

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.

See Also

GMW

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

Generalized Weibull distribution

Description

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

Usage

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

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

qGWF(p, mu, sigma, nu)

rGWF(n, mu, sigma, nu)

hGWF(x, mu, sigma, nu)

Arguments

Details

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

f(x) = \mu \sigma x^{\sigma - 1} \left( 1 - \mu \nu x^\sigma \right)^{\frac{1}{\nu} - 1}

for x > 0, \mu>0, \sigma>0 and -\infty<\nu<\infty.

Value

dGWF gives the density, pGWF gives the distribution function, qGWF gives the quantile function, rGWF generates random deviates and hGWF gives the hazard function.

Author(s)

Jaime Mosquera, jmosquerag@unal.edu.co

References

Mudholkar, G. S., & Kollia, G. D. (1994). Generalized Weibull family: a structural analysis. Communications in statistics-theory and methods, 23(4), 1149-1171.

Examples

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

## The probability density function
curve(
    dGWF(x, mu = 5, sigma = 2, nu = -0.2),
    from = 0, to = 5, col = "red", las = 1, ylab = "f(x)"
)

## The cumulative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(
    pGWF(x, mu = 5, sigma = 2, nu = -0.2),
    from = 0, to = 5, ylim = c(0, 1),
    col = "red", las = 1, ylab = "F(x)"
)
curve(
    pGWF(
        x, mu = 5, sigma = 2, nu = -0.2, 
        lower.tail = FALSE
    ),
    from = 0, to = 5, ylim = c(0, 1),
    col = "red", las = 1, ylab = "R(x)"
)

## The quantile function
p <- seq(from = 0, to = 0.999, length.out = 100)
plot(
    x = qGWF(p, mu = 5, sigma = 2, nu = -0.2),
    y = p, xlab = "Quantile", las = 1,
    ylab = "Probability"
)
curve(
    pGWF(x, mu = 5, sigma = 2, nu = -0.2),
    from = 0, add = TRUE, col = "red"
)

## The random function
hist(
    rGWF(n = 10000, mu = 5, sigma = 2, nu = -0.2),
    freq = FALSE, xlab = "x", las = 1, main = "", ylim = c(0, 2.0)
)
curve(dGWF(x, mu = 5, sigma = 2, nu = -0.2),
    from = 0, add = TRUE, col = "red"
)

## The Hazard function
par(mfrow = c(1, 1))
curve(
    hGWF(x, mu = 0.003, sigma = 5e-6, nu = 0.025),
    from = 0, to = 250, 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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Stacy, E. W. (1962). A generalization of the gamma distribution. The Annals of mathematical statistics, 1187-1192.

See Also

GammaW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Drapella, A. (1993). The complementary Weibull distribution: unknown or just forgotten?. Quality and reliability engineering international, 9(4), 383-385.

See Also

IW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Shahbaz, M. Q., Shahbaz, S., & Butt, N. S. (2012). The Kumaraswamy Inverse Weibull Distribution. Pakistan journal of statistics and operation research, 479-489.

See Also

KumIW

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

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, D. V. (1958). Fiducial distributions and Bayes' theorem. Journal of the Royal Statistical Society. Series B (Methodological), 102-107.

See Also

LIN

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Gumbel, E. J. (1958). Statistics of extremes. Columbia university press.

See Also

LW

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

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

Okasha, H. M., El-Baz, A. H., Tarabia, A. M. K., & Basheer, A. M. (2017). Extended inverse Weibull distribution with reliability application. Journal of the Egyptian Mathematical Society, 25(3), 343-349.

See Also

MOEIW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Ghitany, M. E., Al-Hussaini, E. K., & Al-Jarallah, R. A. (2005). Marshall–Olkin extended Weibull distribution and its application to censored data. Journal of Applied Statistics, 32(10), 1025-1034.

See Also

MOEW

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

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. (2019). The Marshall-Olkin kappa distribution: properties and applications. Journal of King Saud University-Science, 31(4), 684-691.

See Also

MOK

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Lai, C. D., Xie, M., & Murthy, D. N. P. (2003). A modified Weibull distribution. IEEE Transactions on reliability, 52(1), 33-37.

See Also

MW

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 New Exponentiated Exponential distribution

Description

Density, distribution function, quantile function, random generation and hazard function for the two-parameter New Exponentiated Exponential with parameters mu and sigma.

Usage

dNEE(x, mu = 1, sigma = 1, log = FALSE)

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

qNEE(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rNEE(n = 1, mu = 1, sigma = 1)

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

Arguments

Details

The New Exponentiated Exponential distribution with parameters mu and sigma has density given by

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

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

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

Value

dNEE gives the density, pNEE gives the distribution function, qNEE gives the quantile function, rNEE generates random deviates and hNEE gives the hazard function.

Author(s)

Juliana Garcia, juliana.garciav@udea.edu.co

References

Hassan, Anwar, I. H. Dar, and M. A. Lone. "A New Class of Probability Distributions With An Application to Engineering Data." Pakistan Journal of Statistics and Operation Research 20.2 (2024): 217-231.

See Also

NEE

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dNEE(x, mu=0.2, sigma=0.3),
      from=0, to=8, col="cadetblue3", las=1, ylab="f(x)")

curve(dNEE(x, mu=1, sigma=4),
      add=TRUE, col= "purple")

curve(dNEE(x, mu=1.5, sigma=22),
      add=TRUE, col="goldenrod")

curve(dNEE(x, mu=0.5, sigma=2),
      add=TRUE, col="green3")

legend("topright", col=c("cadetblue3", "purple", "goldenrod", "green3"), lty=1, bty="n",
       legend=c("mu=0.2, sigma=0.3",
                "mu=1.0, sigma=4",
                "mu=1.5, sigma=22",
                "mu=0.5, sigma=2"))

# Example 2
# Checking if the cumulative curves converge to 1
curve(pNEE(x, mu=0.2, sigma=0.3), ylim=c(0, 1),
      from=0, to=8, col="cadetblue3", las=1, ylab="F(x)")

curve(pNEE(x, mu=1, sigma=4),
      add=TRUE, col= "purple")

curve(pNEE(x, mu=1.5, sigma=22),
      add=TRUE, col="goldenrod")

curve(pNEE(x, mu=0.5, sigma=2),
      add=TRUE, col="green3")

legend("bottomright", col=c("cadetblue3", "purple", "goldenrod", "green3"), lty=1, bty="n",
       legend=c("mu=0.2, sigma=0.3",
                "mu=1.0, sigma=4",
                "mu=1.5, sigma=22",
                "mu=0.5, sigma=2"))

# Example 3
# Checking the quantile function
mu <- 0.5
sigma <- 2
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qNEE(p, mu=mu, sigma=sigma), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pNEE(x, mu=mu, sigma=sigma), from=0, add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical probabilities
mu <- 0.5
sigma <- 2
x <- rNEE(n=10000, mu=mu, sigma=sigma)
hist(x, freq=FALSE)
curve(dNEE(x, mu=mu, sigma=sigma), col="tomato", add=TRUE)

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

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.

See Also

OW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Ghitany, M. E., Al-Mutairi, D. K., Balakrishnan, N., & Al-Enezi, L. J. (2013). Power Lindley distribution and associated inference. Computational Statistics & Data Analysis, 64, 20-33.

See Also

PL

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

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

Sen, S., Korkmaz, M. Ç., & Yousof, H. M. (2018). The quasi XGamma-Poisson distribution: properties and application. Istatistik Journal of The Turkish Statistical Association, 11(3), 65-76.

See Also

QXGP

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 Reduced New Modified Weibull distribution

Description

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

Usage

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

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

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

rRNMW(n, mu, sigma, nu)

hRNMW(x, mu, sigma, nu)

Arguments

Details

The reduced new modified Weibull with parameters mu, sigma and nu has density given by

f(x) = \frac{1}{2 \sqrt{x}} \left( \mu + \sigma (1 + 2 \nu x) e^{\nu x} \right) e^{-\mu \sqrt{x} - \sigma \sqrt{x} e^{\nu x}}

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

Value

dRNMW gives the density, pRNMW gives the distribution function, qRNMW gives the quantile function, rRNMW generates random deviates and hRNMW gives the hazard function.

Author(s)

Jaime Mosquera, jmosquerag@unal.edu.co

References

Almalki, S. J. (2018). A reduced new modified Weibull distribution. Communications in Statistics-Theory and Methods, 47(10), 2297-2313.

Examples

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

## The probability density function
curve(
    dRNMW(x, mu = 0.05, sigma = 0.00025, nu = 2.2),
    from = 0, to = 5, col = "red", las = 1, ylab = "f(x)"
)

## The cualphalative distribution and the Reliability function
par(mfrow = c(1, 2))
curve(
    pRNMW(x, mu = 0.05, sigma = 0.00025, nu = 2.2),
    from = 0, to = 5, ylim = c(0, 1),
    col = "red", las = 1, ylab = "F(x)"
)
curve(
    pRNMW(
        x, mu = 0.05, sigma = 0.00025, nu = 2.2, 
        lower.tail = FALSE
    ),
    from = 0, to = 5, ylim = c(0, 1),
    col = "red", las = 1, ylab = "R(x)"
)

## The quantile function
p <- seq(from = 0, to = 0.999, length.out = 100)
plot(
    x = qRNMW(p, mu = 0.05, sigma = 0.00025, nu = 2.2),
    y = p, xlab = "Quantile", las = 1,
    ylab = "Probability"
)
curve(
    pRNMW(x, mu = 0.05, sigma = 0.00025, nu = 2.2),
    from = 0, add = TRUE, col = "red"
)

## The random function
hist(
    rRNMW(n = 10000, mu = 0.05, sigma = 0.00025, nu = 2.2),
    freq = FALSE, xlab = "x", las = 1, main = "", ylim = c(0,0.8)
)
curve(dRNMW(x, mu = 0.05, sigma = 0.00025, nu = 2.2),
    from = 0, add = TRUE, col = "red"
)

## The Hazard function
par(mfrow = c(1, 1))
curve(
    hRNMW(x, mu = 0.003, sigma = 5e-6, nu = 0.025),
    from = 0, to = 250, 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

Details

The Reflected Weibull Distribution with parameters mu and sigma has density given by

f(x) = \mu \sigma (-x) ^{\sigma - 1} e^{-\mu(-x)^\sigma},

for x < 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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Cohen, A. C. (1973). The reflected Weibull distribution. Technometrics, 15(4), 867-873.

See Also

RW

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

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, S. J., & Nadarajah, S. (2014). Modifications of the Weibull distribution: A review. Reliability Engineering & System Safety, 124, 32-55.

Sarhan, A. M., & Zaindin, M. (2009). Modified Weibull distribution. APPS. Applied Sciences, 11, 123-136.

See Also

SZMW

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 Wald distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Wald distribution with parameter \mu and \sigma.

Usage

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

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

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

rWALD(n, mu, sigma)

Arguments

Details

The Wald distribution with parameters \mu and \sigma has density given by

f(x |\mu, \sigma)=\frac{\sigma}{\sqrt{2 \pi x^3}} \exp \left[-\frac{(\sigma-\mu x)^2}{2x}\right ],

for x < 0.

Value

dWALD gives the density, pWALD gives the distribution function, qWALD gives the quantile function, rWALD generates random deviates.

Author(s)

Sofia Cuartas, scuartasg@unal.edu.co

References

Heathcote, A. (2004). Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package. Behavior Research Methods, Instruments, & Computers, 36, 678-694.

See Also

WALD

WALD.

Examples

# Example 1
# Plotting the mass function for different parameter values
curve(dWALD(x, mu=1, sigma=1),
      from=0, to=3, col="cadetblue3", las=1, ylab="f(x)")

curve(dWALD(x, mu=1, sigma=2),
      add=TRUE, col= "purple")

curve(dWALD(x, mu=2, sigma=4),
      add=TRUE, col="goldenrod")

legend("topright", col=c("cadetblue3", "purple", "goldenrod"), 
       lty=1, bty="n",
       legend=c("mu=1, sigma=1",
                "mu=1, sigma=2",
                "mu=2, sigma=4"))

# Example 2
# Checking if the cumulative curves converge to 1
curve(pWALD(x, mu=1, sigma=1), ylim=c(0, 1),
      from=0, to=5, col="cadetblue3", las=1, ylab="F(x)")

curve(pWALD(x, mu=1, sigma=2),
      add=TRUE, col= "purple")

curve(pWALD(x, mu=2, sigma=4),
      add=TRUE, col="goldenrod")

legend("bottomright", col=c("cadetblue3", "purple", "goldenrod"), 
       lty=1, bty="n",
       legend=c("mu=1, sigma=1",
                "mu=1, sigma=2",
                "mu=2, sigma=4"))

# Example 3
# Checking the quantile function
mu <- 1
sigma <- 2
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qWALD(p, mu=mu, sigma=sigma), y=p, xlab="Quantile",
     las=1, ylab="Probability")
curve(pWALD(x, mu=mu, sigma=sigma), from=0, add=TRUE, col="red")

# Example 4
# Comparing the random generator output with
# the theoretical probabilities
mu <- 1
sigma <- 20
x <- rWALD(n=10000, mu=mu, sigma=sigma)
hist(x, freq=FALSE)
curve(dWALD(x, mu=mu, sigma=sigma), col="tomato", add=TRUE)

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

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, A. L., & Cordeiro, G. M. (2011). The Weibull-geometric distribution. Journal of Statistical Computation and Simulation, 81(5), 645-657.

See Also

WG

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

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, 1-12.

See Also

WGEE

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

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

Lu, Wanbo, and Daimin Shi. "A new compounding life distribution: the Weibull–Poisson distribution." Journal of applied statistics 39.1 (2012): 21-38.

See Also

WP

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

Auxiliar den_exw function for the Ex-Wald distribution

Description

This function emulates the dExWALD function.

Usage

den_exw(r, m, a, t)

Arguments

Value

returns a vector with probabilities.

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)")

initValuesCJ2

Description

This function generates initial values for CJ2 distribution.

Usage

estim_mu_sigma_CJ2(y)

Arguments

Value

A two-length numeric vector with initial estimates for mu and sigma parameters from CJ2 distribution (see dCJ2).

Initial values for NEE distribution

Description

This function generates initial values for the parameters.

Usage

estim_mu_sigma_NEE(y)

Arguments

Value

returns a vector with the MLE estimations.

Auxiliar function for the Ex-Wald distribution

Description

This function generates start values.

Usage

exwstpt(x, p = 0.5)

Arguments

Value

returns a vector with starting values.

Auxiliar function for the Ex-Wald distribution

Description

This function generates starting values.

Usage

fitexw(rt, p = 0.5, start = exwstpt(rt, p), scaleit = TRUE)

Arguments

Value

returns a vector with cumulative probabilities.

initValuesFWE

Description

This function generates initial values for FWE distribution.

Usage

initValuesFWE(y)

Arguments

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

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.start value for sigma parameter of OW distribution.

• nu.start value for nu parameter of OW distribution.

• sigma.valid search region for sigma parameter of OW distribution.

• nu.valid search region for nu parameter of OW distribution.

• TTTplot Total Time on Test transform computed from the data.

• hazard_type shape 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)

logLik function for CJ2

Description

Calculates logLik for CJ2 distribution.

Usage

logLik_CJ2(logparam = c(0, 0), x)

Arguments

Value

returns the loglikelihood given the parameters and random sample.

logLik function for NEE

Description

Calculates logLik for NEE distribution.

Usage

logLik_NEE(logparam = c(0, 0), x)

Arguments

Value

returns the loglikelihood given the parameters and random sample.

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

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'))

Auxiliar function to obtain minus loglik for Ex-Wald

Description

This function calculates minus loglik.

Usage

negllexw(p, x)

Arguments

Value

returns the minus loglik.

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

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"))
  

Auxiliar pwald function for the Ex-Wald distribution

Description

This function emulates the pWALD function.

Usage

pwald(w, m, a, s = 0)

Arguments

Value

returns a vector with cumulative probabilities.

Auxiliar function to generate random values for Ex-Wald distribution

Description

This function generates another form to generate values.

Usage

rew(x, y)

Arguments

Value

returns a vector with values.

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

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

Auxiliar function for the Ex-Wald distribution

Description

This function is an auxiliar function.

Usage

uandv(
  x,
  y,
  firstblock = 20,
  block = 0,
  tol = .Machine$double.eps^(2/3),
  maxseries = 20
)

Arguments

Value

returns a vector with starting values.

Initial values for WALD

Description

This function generates initial values for the parameters.

Usage

wald_start(y)

Arguments

Value

returns a vector with starting values.