# toppleMeanlink: Link functions for the mean of 1-parameter continuous... In VGAMextra: Additions and Extensions of the 'VGAM' Package

## 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)


### Arguments

 theta Numeric or character. See Links and below for further details. bvalue, inverse, deriv, short, tag See Links.

### 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.

### 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.

### Note

Values of s too close to zero or 1.0 may cause numerical instability. Use argument bvalue to replace them before computing the link.

If theta is character, then arguments inverse and deriv are ignored. See Links for further details.

### Author(s)

V. Miranda and Thomas W. Yee.

topple, Links, newtonRaphson.basic.

### Examples

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

## E2. Some probability link functions ##

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")
abline(v = c(0.5, 1), lty = "dashed")
abline(v = 0, h = 0, lty = "dashed")
legend(0.1, 8,