Package 'RelDists'

Description

The Additive Weibull distribution

Usage

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

Arguments

Details

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

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

for $x > 0$.

Value

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

Author(s)

Amylkar Urrea Montoya, [email protected]

References

Almalki, 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)

Description

The Beta Generalized Exponentiated family

Usage

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

Arguments

Details

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

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

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

Value

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

Author(s)

Johan David Marin Benjumea, [email protected]

References

Almalki, 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"))

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"))

Description

The function BS10() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS10(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where mu represents $\tau$ and sigma represents $\omega$) has density given by

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

for $x>0$, $\mu>0$ and $\sigma>0$. In this parameterization, $E(X) = \frac{\mu^2}{2} + \frac{\mu^4}{8\sigma}$ and $Var(X) = \frac{\mu^6}{8\sigma} + \frac{5\mu^8}{64\sigma^2}$.

Value

Returns a gamlss.family object which can be used to fit a BS10 distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

See Also

dBS10.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(12345)
y <- rBS10(n=100, mu=0.75, sigma=5)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS10)

# 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 ~ BS10
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(0.5 - 1.6 * x1)      # Aprox 0.75
  sigma <- exp(2.2 - 1.2 * x2)   # Aprox 5
  y <- rBS10(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=200)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS10, data=dat)

summary(mod2)

Description

The function BS11() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS11(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where mu represents $\beta$ and sigma represents $\omega$) has density given by

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

for $x>0$, $\mu>0$ and $\sigma>0$. In this parameterization, $E(X) = \mu + \frac{\mu^2}{2\sigma}$ and $Var(X) = \frac{\mu^3}{\sigma} + \frac{5\mu^4}{4\sigma^2}$.

Value

Returns a gamlss.family object which can be used to fit a BS11 distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

See Also

dBS11.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rBS11(n=100, mu=1, sigma=12)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS11)

# 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 ~ BS11
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(0.5 - 1 * x1)      # Aprox 1
  sigma <- exp(1.9 + 1.2 * x2)   # Aprox 12
  y <- rBS11(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=200)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS11, data=dat)

summary(mod2)

Description

The function BS12() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS12(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where mu represents $\beta$ and sigma represents $\psi$) has density given by

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

for $x>0$, $\mu>0$ and $\sigma>0$. In this parameterization, $E(X) = \mu + \frac{\mu}{2\sigma}$ and $Var(X) = \frac{\mu^2}{\sigma} + \frac{5\mu^2}{4\sigma^2}$.

Value

Returns a gamlss.family object which can be used to fit a BS12 distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

See Also

dBS12.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(12345)
y <- rBS12(n=100, mu=1, sigma=30)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS12)

# 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 ~ BS12
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(0.5 - 1 * x1)      # Aprox 1
  sigma <- exp(2.2 + 2.4 * x2)   # Aprox 30
  y <- rBS12(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=200)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS12, data=dat)

summary(mod2)

Description

The function BS13() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS13(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where mu represents $\omega$ and sigma represents $\psi$) has density given by

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

for $x>0$, $\mu>0$ and $\sigma>0$. In this parameterization, $E(X) = \frac{\mu}{\sigma} + \frac{\mu}{2\sigma^2}$ and $Var(X) = \frac{\mu^2}{\sigma^3} + \frac{5\mu^2}{4\sigma^4}$.

Value

Returns a gamlss.family object which can be used to fit a BS13 distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

See Also

dBS13.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123456)
y <- rBS13(n=500, mu=5, sigma=30)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS13,
               control=gamlss.control(n.cyc=300, trace=TRUE))

# 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 ~ BS13
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.1 + 1.1 * x1)      # Aprox 5
  sigma <- exp(2.2 + 2.4 * x2)   # Aprox 30
  y <- rBS13(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=200)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS13, data=dat,
               control=gamlss.control(n.cyc=500, trace=TRUE))

summary(mod2)

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"))

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)

Description

The function BS4() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS4(mu.link = "log", sigma.link = "log")

Arguments

Details

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

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

for $x>0$, $\mu>0$ and $\sigma>0$. In this parameterization, $E(X) = \frac{\sigma \mu + 1/2}{\mu^2}$ and $Var(X) = \frac{\sigma \mu + 5/4}{\mu^4}$.

Value

Returns a gamlss.family object which can be used to fit a BS4 distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Ahmed, S. E., Budsaba, K., Lisawadi, S., & Volodin, A. (2008). Parametric estimation for the Birnbaum-Saunders lifetime distribution based on a new parametrization. Thailand Statistician, 6(2), 213-240.

See Also

dBS4.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123)
y <- rBS4(n=50, mu=2, sigma=0.2)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS4)

# 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 ~ BS4
## Not run: 
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.45 - 3 * x1)
  sigma <- exp(2 - 1.5 * x2)
  y <- rBS4(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=BS4, data=dat,
               control=gamlss.control(n.cyc=100))

summary(mod2)

## End(Not run)

# Example 3
# Taken from Ahmed et al. (2008) on page 227
# The response variable is the fatigue
# life of 6061-T6 aluminum coupons.
## Not run: 
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, 130, 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, family=BS4, 
               control=gamlss.control(n.cyc=3000))
  
summary(mod3)
exp(coef(mod3, what="mu"))
exp(coef(mod3, what="sigma"))

plot(density(y))
curve(dBS4(x, mu=0.5145121, sigma=67.81761),
      add=TRUE, col="tomato", lwd=2)
legend("topright", legend=c("Empirical", "Estimated"), 
       col=c("black", "tomato"), lty=1)

## End(Not run)

# Example 4
# Taken from Ahmed et al. (2008) on page 228
# The response variable is the fatigue life in
# hours of 10 bearings of a certain type. 
## Not run: 
y <- c(152.7, 172.0, 172.5, 173.5, 193.0,
       204.7, 216.5, 234.9, 262.6, 422.6)

mod4 <- gamlss(y~1, family=BS4, 
               control=gamlss.control(n.cyc=3000))

summary(mod4)
exp(coef(mod4, what="mu"))
exp(coef(mod4, what="sigma"))

## End(Not run)

Description

The function BS5() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS5(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where sigma represents the precision parameter $\delta$) has density given by

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

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

Value

Returns a gamlss.family object which can be used to fit a BS5 distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

See Also

dBS5.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(123)
y <- rBS5(n=50, mu=1, sigma=25)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS5)

# 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 ~ BS5
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.5 - 3 * x1)        # Aprox 1
  sigma <- exp(2.4 + 1.7 * x2)   # Aprox 25
  y <- rBS5(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

dat <- gendat(n=100)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS5, data=dat)

summary(mod2)

Description

The function BS6() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS6(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where mu represents the true mean and sigma represents the shape parameter $\alpha$) has density given by

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

for $x>0$, $\mu>0$ and $\sigma>0$. In this parameterization, $E(X) = \mu$ and $Var(X) = [\mu\sigma]^2 \left[ \frac{4+5\sigma^2}{(2+\sigma^2)^2} \right]$.

Value

Returns a gamlss.family object which can be used to fit a BS6 distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

See Also

dBS6.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(1234)
y <- rBS6(n=50, mu=1, sigma=0.1)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS6)

# 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 ~ BS6
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.5 - 3 * x1)        # Aprox 1
  sigma <- exp(0.5 - 3.5 * x2)   # Aprox 0.1
  y <- rBS6(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

dat <- gendat(n=100)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS6, data=dat)

summary(mod2)

Description

The function BS7() defines the Birnbaum-Saunders distribution, a two-parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

Usage

BS7(mu.link = "log", sigma.link = "log")

Arguments

Details

The Birnbaum-Saunders distribution with parameters mu and sigma (where mu represents the true variance $\sigma^2$ and sigma represents the shape parameter $\alpha$) has density given by

$f(x|\mu,\sigma) = \frac{1}{\sqrt{2\pi}} \exp\left( -\frac{1}{2\mu^2} \left[ \frac{\mu\sqrt{4+5\mu^2}}{2\sqrt{\sigma}x^{-1}} + \frac{2\sqrt{\sigma}\{x\mu\}^{-1}}{\sqrt{4+5\mu^2}} - 2 \right] \right) \times \left[ \frac{\{x\mu\}^{-1/2}\{4+5\mu^2\}^{1/4}}{2^{3/2}\sigma^{1/4}} + \frac{\sigma^{1/4}}{\{x\mu\}^{3/2}\sqrt{2}\{4+5\mu^2\}^{1/4}} \right]$

for $x>0$, $\mu>0$ and $\sigma>0$. In this parameterization, $E(X) = \frac{[2+\mu^2]\sqrt{\sigma}}{\mu\sqrt{4+5\mu^2}}$ and $Var(X) = \sigma$.

Value

Returns a gamlss.family object which can be used to fit a BS7 distribution in the gamlss() function.

Author(s)

David Villegas Ceballos, [email protected]

References

Santos-Neto, M., Cysneiros, F. J. A., Leiva, V., & Ahmed, S. E. (2012). On new parameterizations of the Birnbaum-Saunders distribution. Pakistan Journal of Statistics, 28(1), 1-26.

See Also

dBS7.

Examples

# Example 1
# Generating some random values with
# known mu and sigma
set.seed(12345)
y <- rBS7(n=100, mu=0.2, sigma=10)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS7, 
               control=gamlss.control(n.cyc=1000))

# 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 ~ BS7
## Not run: 
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(0.6 - 4.4 * x1)      # Aprox 0.2
  sigma <- exp(1.6 + 1.5 * x2)   # Aprox 10
  y <- rBS7(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(123)
dat <- gendat(n=200)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS7, data=dat,
               control=gamlss.control(n.cyc=1000))

summary(mod2)

## End(Not run)

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, [email protected]

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)

Description

The Cosine Sine Exponential family

Usage

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

Arguments

Details

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

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

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

Value

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

Author(s)

Johan David Marin Benjumea, [email protected]

References

Chesneau, C., Bakouch, 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"))

Description

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

Usage

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

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

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

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

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

Arguments

Details

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

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

for $x > 0$.

Value

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

Author(s)

Amylkar Urrea Montoya, [email protected]

References

Almalki, 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

Description

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

Usage

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

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

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

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

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

Arguments

Details

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

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

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

Value

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

Author(s)

Johan David Marin Benjumea, [email protected]

References

Almalki, 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

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)

# 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)