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The Birnbaum-Saunders family - Santos-Neto et al. (2012) (P11 Based on the fourth Tweedie)

Source: R/BS13.R
BS13.Rd

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

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

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 BS13 distribution in the gamlss() function.

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 \(Var(X) = \frac{\mu^2}{\sigma^3} + \frac{5\mu^2}{4\sigma^4}\).

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.

Author

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

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))
#> GAMLSS-RS iteration 1: Global Deviance = -959.7991 
#> GAMLSS-RS iteration 2: Global Deviance = -1094.719 
#> GAMLSS-RS iteration 3: Global Deviance = -1201.023 
#> GAMLSS-RS iteration 4: Global Deviance = -1287.423 
#> GAMLSS-RS iteration 5: Global Deviance = -1359.318 
#> GAMLSS-RS iteration 6: Global Deviance = -1420.617 
#> GAMLSS-RS iteration 7: Global Deviance = -1473.785 
#> GAMLSS-RS iteration 8: Global Deviance = -1520.15 
#> GAMLSS-RS iteration 9: Global Deviance = -1562.535 
#> GAMLSS-RS iteration 10: Global Deviance = -1598.916 
#> GAMLSS-RS iteration 11: Global Deviance = -1631.619 
#> GAMLSS-RS iteration 12: Global Deviance = -1661.323 
#> GAMLSS-RS iteration 13: Global Deviance = -1688.216 
#> GAMLSS-RS iteration 14: Global Deviance = -1712.455 
#> GAMLSS-RS iteration 15: Global Deviance = -1734.883 
#> GAMLSS-RS iteration 16: Global Deviance = -1755.712 
#> GAMLSS-RS iteration 17: Global Deviance = -1774.794 
#> GAMLSS-RS iteration 18: Global Deviance = -1792.531 
#> GAMLSS-RS iteration 19: Global Deviance = -1803.669 
#> GAMLSS-RS iteration 20: Global Deviance = -1818.839 
#> GAMLSS-RS iteration 21: Global Deviance = -1833.52 
#> GAMLSS-RS iteration 22: Global Deviance = -1847.151 
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#> GAMLSS-RS iteration 48: Global Deviance = -2021.02 
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# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
#> (Intercept) 
#>    5.239938 
exp(coef(mod1, what="sigma"))
#> (Intercept) 
#>    31.57119 

# 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))
#> GAMLSS-RS iteration 1: Global Deviance = -354.099 
#> GAMLSS-RS iteration 2: Global Deviance = -353.136 
#> GAMLSS-RS iteration 3: Global Deviance = -451.8645 
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#> GAMLSS-RS iteration 301: Global Deviance = -794.9821 
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#> GAMLSS-RS iteration 309: Global Deviance = -794.9909 

summary(mod2)
#> Warning: summary: vcov has failed, option qr is used instead
#> ******************************************************************
#> Family:  c("BS13", "Birnbaum-Saunders - Thirteenth parameterization") 
#> 
#> Call:  gamlss(formula = y ~ x1, sigma.formula = ~x2, family = BS13,  
#>     data = dat, control = gamlss.control(n.cyc = 500, trace = TRUE)) 
#> 
#> Fitting method: RS() 
#> 
#> ------------------------------------------------------------------
#> Mu link function:  log
#> Mu Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  1.12047    0.02428   46.16   <2e-16 ***
#> x1           1.05391    0.04389   24.01   <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)  2.15332    0.03189   67.53   <2e-16 ***
#> x2           2.47011    0.04375   56.45   <2e-16 ***
#> ---
#> 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:  309 
#>  
#> Global Deviance:     -794.9909 
#>             AIC:     -786.9909 
#>             SBC:     -773.7976 
#> ******************************************************************