RelDists (version 1.0.2)

BS4: The Birnbaum-Saunders family - Ahmed et al. (2008)

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")

Value

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

Arguments

mu.link

defines the mu.link, with "log" link as the default for the mu parameter.

sigma.link

defines the sigma.link, with "log" link as the default for the sigma parameter.

Author

David Villegas Ceballos,

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 \(Var(X) = \frac{\sigma \mu + 5/4}{\mu^4}\).

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
if (FALSE) {
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)
}

# Example 3
# Taken from Ahmed et al. (2008) on page 227
# The response variable is the fatigue
# life of 6061-T6 aluminum coupons.
if (FALSE) {
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)
}

# 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. 
if (FALSE) {
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"))
}