toppleMlink | R Documentation |

Computes the `toppleMlink`

transformation, its inverse and
the first two derivatives.

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

`theta` |
Numeric or character.
See |

`bvalue, inverse, deriv, short, tag` |
See |

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`

.

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`

.

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.

V. Miranda and Thomas W. Yee.

`topple`

,
`Links`

,
`newtonRaphson.basic`

.

```
## 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 ##
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)
```

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