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The Birnbaum-Saunders family - Santos-Neto et al. (2012) (P5 Based on the variance)

Source: R/BS7.R
BS7.Rd

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

defines the mu.link, with "log" link as the default for the mu parameter (representing the variance).

defines the sigma.link, with "log" link as the default for the sigma parameter (representing the shape).

Value

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

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 \(Var(X) = \sigma\).

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.

Author

David Villegas Ceballos, david.villegas1@udea.edu.co

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))
#> GAMLSS-RS iteration 1: Global Deviance = 523.2187 
#> GAMLSS-RS iteration 2: Global Deviance = 523.2067 
#> GAMLSS-RS iteration 3: Global Deviance = 523.1963 
#> GAMLSS-RS iteration 4: Global Deviance = 523.1875 
#> GAMLSS-RS iteration 5: Global Deviance = 523.1799 
#> GAMLSS-RS iteration 6: Global Deviance = 523.1734 
#> GAMLSS-RS iteration 7: Global Deviance = 523.1679 
#> GAMLSS-RS iteration 8: Global Deviance = 523.1631 
#> GAMLSS-RS iteration 9: Global Deviance = 523.159 
#> GAMLSS-RS iteration 10: Global Deviance = 523.1555 
#> GAMLSS-RS iteration 11: Global Deviance = 523.1525 
#> GAMLSS-RS iteration 12: Global Deviance = 523.1499 
#> GAMLSS-RS iteration 13: Global Deviance = 523.1477 
#> GAMLSS-RS iteration 14: Global Deviance = 523.146 
#> GAMLSS-RS iteration 15: Global Deviance = 523.1445 
#> GAMLSS-RS iteration 16: Global Deviance = 523.1433 
#> GAMLSS-RS iteration 17: Global Deviance = 523.1421 
#> GAMLSS-RS iteration 18: Global Deviance = 523.1412 

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
#> (Intercept) 
#>    0.214036 
exp(coef(mod1, what="sigma"))
#> (Intercept) 
#>    11.56968 

# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ BS7
if (FALSE) { # \dontrun{
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)
} # }