The log-EVBS distribution for GAMLSS models

Defines the log-Extreme-Value Birnbaum-Saunders distribution as a three-parameter gamlss family, so that each parameter may depend on covariates through a linear predictor or a smooth term. This is the natural way to let the tail-shape parameter vary with covariates, which the fixed three-parameter fit of evbsreg.fit does not allow.

Usage

logEVBS(mu.link = "identity", sigma.link = "log", nu.link = "identity")

Arguments

mu.link

Link for the location parameter \(\mu\) (default "identity"; the predictor is on the log-response scale).

sigma.link

Link for the scale parameter \(\alpha > 0\) (default "log").

nu.link

Link for the extreme-value shape parameter \(\gamma\) (default "identity").

Value

A gamlss.family object.

Details

The response is \(Y = \log T\) where \(T\) follows the EVBS distribution, so that \(Z = (2/\sigma)\sinh\{(Y-\mu)/2\} \sim \mathrm{GEV}(0,1,\nu)\). The parameter map is \(\mu = \eta = x^\top\beta\) (identity link), \(\sigma = \alpha\) (log link, positive) and \(\nu = \gamma\) (identity link, real-valued).

With a constant predictor the fitted \((\mu, \sigma, \nu)\) reproduce the \((\eta, \alpha, \gamma)\) of evbsreg.fit; this equivalence is the recommended check after loading the family. The score and second-order derivatives required by gamlss are obtained by automatic numerical differentiation of the analytic log-density, which is exact to working precision and avoids transcription error in the lengthy closed forms.

Examples

if (FALSE) { # \dontrun{
  library(gamlss)
  data(itajai)
  y <- log(itajai$wind)
  # constant model: should match evbsreg.fit on the log scale
  m0 <- gamlss(y ~ 1, family = logEVBS())
  # tail shape depending on pressure:
  m1 <- gamlss(y ~ pressure, nu.formula = ~ pressure,
               data = itajai, family = logEVBS())
} # }