VGAMextra (version 0.0-6)

toppleMlink: Link functions for the mean of 1--parameter continuous distribution: The Topp--Leone distribution.

Description

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

Usage

toppleMlink(theta, bvalue = NULL, inverse = FALSE,
                  deriv = 0, short = TRUE, tag = FALSE)

Value

For deriv = 0, the toppleMlink transformation of

theta when inverse = FALSE. If inverse = TRUE, then theta becomes \(\eta\), and the inverse transformation is required. However, it can't be expressed in close form. Therefore, the approximate

inverse image of entered theta computed by

newtonRaphson.basic

is returned.

For deriv = 1,

\(d\)

eta / \(d\)

theta when inverse = FALSE. If inverse = TRUE, then

\(d\)

theta / \(d\)

eta as a function of

theta.

Arguments

theta

Numeric or character. See Links and below for further details.

bvalue, inverse, deriv, short, tag

See Links.

Author

V. Miranda and Thomas W. Yee.

Details

The toppleMlink transformation arises as a link function to model the mean of the Topp--Leone distribution, topple. It is defined as

$$ \eta = {\tt{logit}} \left( \left( 1 - \frac{4^{s} \Gamma(1 + s)^2}{ \Gamma(2 + 2s)} \right) / sup.tp \right).$$

Here, \(0 < s < 1\) is a shape parameter as in topple, whereas \(sup.tp\) is the supremum of $$ 1 - \frac{4^{s} \Gamma(1 + s)^2}{ \Gamma(2 + 2s)}, $$ in \((0, 1)\), as a function of \(s\).

For numerical values of \(s\) out of \((0, 1)\), this link may result in Inf, -Inf, NA or NaN.

See Also

topple, Links, newtonRaphson.basic.

Examples

 ## E1. The toppleMlink() and its inverse ##
    theta <- ppoints(10)
    eta   <- toppleMlink(toppleMlink(theta = theta), inverse  =TRUE)
    summary(eta - theta)     # Zero


  ## E2. Some probability link functions ##
  # \donttest{
my.probs <- ppoints(100)

par(lwd = 2)
plot(my.probs, logitlink(my.probs), xlim = c(-0.1, 1.1), ylim = c(-5, 8),
     type = "l", col = "limegreen", 
     ylab = "transformation", las = 1, main = "Some probability link functions")
lines(my.probs, geometricffMlink(my.probs), col = "gray50")
lines(my.probs, logffMlink(my.probs), col = "blue")
lines(my.probs, probitlink(my.probs), col = "purple")
lines(my.probs, clogloglink(my.probs), col = "chocolate")
lines(my.probs, cauchitlink(my.probs), col = "tan")
lines(my.probs, toppleMlink(my.probs), col = "black")
abline(v = c(0.5, 1), lty = "dashed")
abline(v = 0, h = 0, lty = "dashed")
legend(0.1, 8, 
      c( "toppleMlink", "geometricffMlink", "logffMlink",
      "logitlink", "probitlink", 
      "clogloglink", "cauchitlink"), 
      col = c("black", "gray50", "blue", "limegreen", "purple", "chocolate", "tan"),
      lwd = 1, cex = 0.5)
par(lwd = 1) 
 # }