VGAMextra (version 0.0-1)

gamma1Qlink: Link functions for the quantiles of several 1--parameter continuous distributions

Description

Computes the gamma1Qlink transformation, its inverse and the first two derivatives.

Usage

gamma1Qlink(theta, p = stop("Argument 'p' must be specified."),
               bvalue = NULL, inverse = FALSE,
               deriv = 0, short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. It is \(\theta\) by default although it could be \(\eta\) depending upon other arguments. See Links for further details about this.

p

A numeric vector of \(p\)--quantiles (numbers between 0 and 1) to be modeled by this link function.

bvalue, inverse, deriv, short, tag

See Links.

Value

For deriv = 0, the gamma1Qlink transformation of theta, when inverse = FALSE. If inverse = TRUE, then theta becomes \(\eta\), and therefore, the approximate inverse image of \(\eta\) is returned.

For deriv = 1, the partial derivative \(d\) eta / \(d\) theta is returned, if inverse = FALSE. If inverse = TRUE, then the reciprocal \(d\) theta / \(d\) eta as a function of theta.

If deriv = 2, then the second order derivatives as a function of theta.

Details

This link function has been specifically designed to model any \(p\)--quantile of the 1--parameter gamma distribution, gamma1, in the VGLM/VGAM context. It is defined as $$\eta = \log {\tt{qgamma}}({\tt{p}}, {\tt{shape =}} s),$$ where \(s\) is a positive shape parameter as in gamma1, whilst \({\tt{qgamma()}}\) is the quantile function qgamma.

The inverse of the gamma1Qlink cannot be expressed in closed form. Instead, the inverse image, \(s_{\eta}\), of \(\eta\) is numerically approximated by newtonRaphson.basic.

Numerical values of \(s\) or \(p\) out of range will result in Inf, -Inf, NA or NaN correspondingly.

Arguments inverse and deriv are dismissed if theta is character.

See Also

gamma1, qgamma, Links.

Examples

# NOT RUN {
  ## E1. gamma1QLink() and values causing NaNs or out of range  ##
  
  p <- 0.75                            # The third quartile is of interest.
  my.s <- seq(0, 5, by = 0.1)[-1]
  
  max(my.s - gamma1Qlink(gamma1Qlink(my.s, p = p), p = p, inverse  =TRUE)) ## Zero
  
  ## E2. Special values of theta ##
  gamma1Qlink(theta = c(-0.15, -0.10, 0, 1:10) , p = p, inverse  = FALSE)  ## NaNs
  gamma1Qlink(theta = c(-5, -3, 0, 1:10) , p = p, inverse  = TRUE)         ## Out of range
   
   
  ## E3. Plot of gamma1QLink() and its inverse. ##
  
# }
# NOT RUN {
  
    # gamma1Qlink()
    plot(gamma1Qlink(theta = my.s, p = p) ~ my.s,
         type = "l", col = "blue", lty = "dotted", lwd = 3,
         xlim = c(-0.1, 5), ylim = c(-5, 15), las = 1,
         main = c("Blue is gamma1Qlink(), green is the inverse"),
         ylab = "gamma1Qlink transformation", xlab = "theta")
     abline(h = 0, v = 0, lwd = 2)

     # The inverse
     lines(my.s, gamma1Qlink(theta = my.s, p = p, inverse = TRUE),
           col = "green", lwd = 2, lty = "dashed")
           
      # The identity function, for double-checking.
     lines(my.s, my.s, lty = "dotted")
  
# }