RelDists (version 1.0.1)

BS3: 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")

Value

Returns a gamlss.family object which can be used to fit a BS3 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 "logit" link as the default for the sigma.

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 \(Var(X)=(\mu\sigma)^2(1+5\sigma^2/4)\).

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
if (FALSE) {
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)
}

# 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)