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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