The Birnbaum-Saunders family - Santos-Neto et al. (2012) (P10 Based on the third Tweedie)

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

mu.link: defines the mu.link, with "log" link as the default for the mu parameter (representing the scale \(\beta\)).
sigma.link: defines the sigma.link, with "log" link as the default for the sigma parameter (representing the shape \(\psi\)).

Value

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

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 \(Var(X) = \frac{\mu^2}{\sigma} + \frac{5\mu^2}{4\sigma^2}\).

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.

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 <- rBS12(n=100, mu=1, sigma=30)

# Fitting the model
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS12)
#> GAMLSS-RS iteration 1: Global Deviance = -42.2156 
#> GAMLSS-RS iteration 2: Global Deviance = -42.2157 

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
#> (Intercept) 
#>    1.001539 
exp(coef(mod1, what="sigma"))
#> (Intercept) 
#>     25.8992 

# 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)
#> GAMLSS-RS iteration 1: Global Deviance = -108.1624 
#> GAMLSS-RS iteration 2: Global Deviance = -109.0761 
#> GAMLSS-RS iteration 3: Global Deviance = -109.0766 

summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family:  c("BS12", "Birnbaum-Saunders - Twelfth parameterization") 
#> 
#> Call:  gamlss(formula = y ~ x1, sigma.formula = ~x2, family = BS12,  
#>     data = dat) 
#> 
#> Fitting method: RS() 
#> 
#> ------------------------------------------------------------------
#> Mu link function:  log
#> Mu Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   0.5157     0.0228   22.62   <2e-16 ***
#> x1           -1.0409     0.0411  -25.33   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> ------------------------------------------------------------------
#> Sigma link function:  log
#> Sigma Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)   2.0467     0.2020  10.132  < 2e-16 ***
#> x2            2.7063     0.3582   7.554 1.52e-12 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> ------------------------------------------------------------------
#> No. of observations in the fit:  200 
#> Degrees of Freedom for the fit:  4
#>       Residual Deg. of Freedom:  196 
#>                       at cycle:  3 
#>  
#> Global Deviance:     -109.0766 
#>             AIC:     -101.0766 
#>             SBC:     -87.88336 
#> ******************************************************************