VGAMextra (version 0.0-1)

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

Description

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

Usage

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

Arguments

theta

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

bvalue, inverse, deriv, short, tag

See Links.

Value

For deriv = 0, the toppleMeanlink 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.

Details

The toppleMeanlink 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

# NOT RUN {
 ## E1. The toppleMeanlink() and its inverse ##
    theta <- ppoints(10)
    eta   <- toppleMeanlink(toppleMeanlink(theta = theta), inverse  =TRUE)
    summary(eta - theta)     # Zero


  ## E2. Some probability link functions ##
  
# }
# NOT RUN {
my.probs <- ppoints(100)

par(lwd = 2)
plot(my.probs, logit(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, geometricffMeanlink(my.probs), col = "gray50")
lines(my.probs, logffMeanlink(my.probs), col = "blue")
lines(my.probs, probit(my.probs), col = "purple")
lines(my.probs, cloglog(my.probs), col = "chocolate")
lines(my.probs, cauchit(my.probs), col = "tan")
lines(my.probs, toppleMeanlink(my.probs), col = "black")
abline(v = c(0.5, 1), lty = "dashed")
abline(v = 0, h = 0, lty = "dashed")
legend(0.1, 8, 
      c( "toppleMeanlink", "geometricffMeanlink", "logffMeanlink", "logit", "probit", 
      "cloglog", "cauchit"), 
      col = c("black", "gray50", "blue", "limegreen", "purple", "chocolate", "tan"),
      lwd = 1, cex = 0.5)
par(lwd = 1) 
 
# }
# NOT RUN {
# }