benini1Qlink transformation, its inverse and the
first two derivatives.
1 2 3 4
Numeric or character. See below for further details.
Numeric. A single value between 0.0 and 1.0. It is the p–quantile to be modeled by this link function.
This is a link function to model any p–quantile of the
1–parameter Benini distribution. It is called the
benini1Qlink transformation defined as
log y0 + ( -log(1 - p)/s )^(1/2),
where y0 > 0 is a scale parameter and s is
a positive shape parameter, as in
Numerical values of s or p out of range may
In particular, arguments
theta is character.
deriv = 0, the
benini1Qlink transformation of
theta, when inverse = FALSE. If
inverse = TRUE, then
the inverse transformation given by
-log(1 - p) / (theta - log y0)^2 is returned.
deriv = 1, this function returns the derivative
eta / d
inverse = FALSE. Else, the reciprocal
theta / d
eta as a function of
If deriv = 2, then the second order derivatives in terms of
theta are accordingly returned.
The horizontal straight line log y0 is a lower asymptote
for this link function as theta increases to ∞.
inverse = TRUE and
deriv = 0 entries at
theta becoming η must be greater than
log y0. Else,
NaN will be returned.
See examples 2 and 3 below.
Numerical instability may occur for values
theta too close
to zero or lower than \log y0.
bvalue to replace them before computing the link.
V. Miranda and Thomas W. Yee.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
## E1. benini1Qlink() and its inverse ## p <- 0.50; y0 = 1.25 ## Modeling the median my.s <- seq(0, 5, by = 0.1)[-1] max(my.s - benini1Qlink(benini1Qlink(my.s, p = p, y0 = y0), p = p, y0 = y0, inverse =TRUE)) ## Zero ## E2. Plot of the benini1Qlink() transformation and its inverse ## ## Note, inverse = TRUE implies that argument 'theta' becomes 'eta'. ## ## which must be greater than log(y0). Else, value less than log(y0) ## ## are replaced by NaN. ## #--- THE LINK my.b <- seq(0, 5, by = 0.01)[-1] plot(benini1Qlink(theta = my.b, p = p, y0 = y0) ~ my.b, type = "l", col = "blue", lty = "dotted", lwd = 3, xlim = c(-0.1, 6), ylim = c(-0.1, 5), las = 1, main = c("Blue is benini1Qlink(), green is the inverse"), ylab = "eta = benini1Qlink", xlab = "theta") abline(h = 0, v = 0, lwd = 2) #--- THE INVERSE lines(my.b, benini1Qlink(theta = my.b, p = p, y0 = y0, inv = TRUE), col = "green", lwd = 2, lty = "dashed") #--- Tracing the identity function for double--check lines(my.b, my.b) ## E3. WARNING! The first two values are less than log(y0) ## benini1Qlink(theta = c(0.10, 0.15, 0.25, 0.35) , p = p, y0 = y0, inverse = TRUE)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.