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The Birnbaum-Saunders family - Santos-Neto et al. (2012) (P6 Based on the variance 2)

Source: R/BS8.R
BS8.Rd

The function BS8() 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

BS8(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 BS8 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

\(f(x|\mu,\sigma) = \frac{\sqrt{\mu}} {2\sqrt{2\pi\sigma}} \left[ \left\{ \frac{1}{2x} \sqrt{\frac{5\sigma}{\mu(\mu-1)}} \right\}^{1/2} + \left\{ \frac{1}{2x} \sqrt{\frac{5\sigma}{\mu(\mu-1)}} \right\}^{3/2} \right] \exp\left( -\frac{5}{8(\mu-1)} \left[ \frac{2x\sqrt{\mu(\mu-1)}}{\sqrt{5\sigma}} + \frac{\sqrt{5\sigma}} {2+\sqrt{\mu(\mu-1)}} -2 \right] \right) \)

for \(x>0\), \(\mu>1\) and \(\sigma>0\). In this parameterization, \(E(X) = \frac{[2\mu+3]\sqrt{\sigma}}{\sqrt{20\mu(\mu-1)}}\) 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

dBS8.

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 <- rBS8(n=10, mu=1.1, sigma=10)

# Fitting the model using default link function
require(gamlss)
mod1 <- gamlss(y~1, sigma.fo=~1, family=BS8,
               control=gamlss.control(n.cyc=1000))
#> GAMLSS-RS iteration 1: Global Deviance = 50.8552 
#> GAMLSS-RS iteration 2: Global Deviance = 50.8413 
#> GAMLSS-RS iteration 3: Global Deviance = 50.8295 
#> GAMLSS-RS iteration 4: Global Deviance = 50.8196 
#> GAMLSS-RS iteration 5: Global Deviance = 50.8113 
#> GAMLSS-RS iteration 6: Global Deviance = 50.8044 
#> GAMLSS-RS iteration 7: Global Deviance = 50.7986 
#> GAMLSS-RS iteration 8: Global Deviance = 50.7938 
#> GAMLSS-RS iteration 9: Global Deviance = 50.7899 
#> GAMLSS-RS iteration 10: Global Deviance = 50.7866 
#> GAMLSS-RS iteration 11: Global Deviance = 50.7839 
#> GAMLSS-RS iteration 12: Global Deviance = 50.7817 
#> GAMLSS-RS iteration 13: Global Deviance = 50.7794 
#> GAMLSS-RS iteration 14: Global Deviance = 50.7776 
#> GAMLSS-RS iteration 15: Global Deviance = 50.7762 
#> GAMLSS-RS iteration 16: Global Deviance = 50.7751 
#> GAMLSS-RS iteration 17: Global Deviance = 50.7742 

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod1, what="mu"))
#> (Intercept) 
#>    1.080822 
exp(coef(mod1, what="sigma"))
#> (Intercept) 
#>    10.56283 

# Fitting again the model using our own link function.
# As mu > 1, we can use
#     link function:    g(mu) = log(mu - 1)
# inv link function: g-1(eta) = exp(eta) + 1

# The following function "log_1_to_inf" is defined with the informatio
# of the link and inv link functions:

if (FALSE) { # \dontrun{

log_1_to_inf <- function() {
  linkfun <- function(mu) { log(mu - 1) }
  linkinv <- function(eta) { 
    thresh <- log(.Machine$double.xmax)
    eta <- pmin(thresh, pmax(eta, -700))
    1 + exp(eta) 
  }
  mu.eta <- function(eta) { 
    thresh <- log(.Machine$double.xmax)
    eta <- pmin(thresh, pmax(eta, -700))
    pmax(exp(eta), .Machine$double.eps)
  }
  valideta <- function(eta) { TRUE }
  link <- "log_1_to_inf"
  structure(list(linkfun = linkfun, linkinv = linkinv, mu.eta = mu.eta,
                 valideta = valideta, name = link), class = "link-gamlss")
}

mod99 <- gamlss(y~1, sigma.fo=~1, 
                family=BS8(mu.link = log_1_to_inf()),
                control=gamlss.control(n.cyc=1000))

# Extracting the fitted values for mu and sigma
# using the inverse link function
exp(coef(mod99, what="mu")) + 1
exp(coef(mod99, what="sigma"))

# Example 2
# Generating random values for a regression model

# A function to simulate a data set with Y ~ BS8
gendat <- function(n) {
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.6 - 2.4 * x1) + 1  # Aprox 2.45 with link function log(mu - 1)
  sigma <- exp(1.6 + 1.5 * x2)   # Aprox 10 with link function log(sigma)
  y <- rBS8(n=n, mu=mu, sigma=sigma)
  data.frame(y=y, x1=x1, x2=x2)
}

set.seed(1234)
dat <- gendat(n=400)

mod2 <- gamlss(y~x1, sigma.fo=~x2, 
               family=BS8(mu.link = log_1_to_inf()), 
               data=dat,
               control=gamlss.control(n.cyc=100))

summary(mod2)

} # }