Calculate or plot the dependence function \(A\) for nine parametric bivariate extreme value models.
abvevd(x = 0.5, dep, asy = c(1,1), alpha, beta, model = c("log", "alog",
"hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"),
rev = FALSE, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1,
blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "t",
ylab = "A(t)", ...)
abvevd
calculates or plots the dependence function
for one of nine parametric bivariate extreme value models,
at specified parameter values.
A vector of values at which the dependence function is
evaluated (ignored if plot or add is TRUE
). \(A(1/2)\)
is returned by default since it is often a useful summary of
dependence.
Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.
A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.
Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.
The specified model; a character string. Must be
either "log"
(the default), "alog"
, "hr"
,
"neglog"
, "aneglog"
, "bilog"
,
"negbilog"
, "ct"
or "amix"
(or any unique
partial match), for the logistic, asymmetric logistic,
Husler-Reiss, negative logistic, asymmetric negative logistic,
bilogistic, negative bilogistic, Coles-Tawn and asymmetric
mixed models respectively. The definition of each model is given
in rbvevd
. If parameter arguments are given that do
not correspond to the specified model those arguments are
ignored, with a warning.
Logical; reverse the dependence function? This is
equivalent to evaluating the function at 1-x
.
Logical; if TRUE
the function is plotted. The
x and y values used to create the plot are returned invisibly.
If plot
and add
are FALSE
(the default),
the arguments following add
are ignored.
Logical; add to an existing plot? The existing plot
should have been created using either abvevd
or
abvnonpar
, the latter of which plots (or calculates)
a non-parametric estimate of the dependence function.
Function and border line types. Set blty
to zero to omit the border.
Function an border line widths.
Line colour.
x and y-axis limits.
x and y-axis labels.
Other high-level graphics parameters to be passed to
plot
.
Any bivariate extreme value distribution can be written as $$G(z_1,z_2) = \exp\left[-(y_1+y_2)A\left( \frac{y_1}{y_1+y_2}\right)\right]$$ for some function \(A(\cdot)\) defined on \([0,1]\), where $$y_i = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}$$ for \(1+s_i(z_i-a_i)/b_i > 0\) and \(i = 1,2\), with the (generalized extreme value) marginal parameters given by \((a_i,b_i,s_i)\), \(b_i > 0\). If \(s_i = 0\) then \(y_i\) is defined by continuity.
\(A(\cdot)\) is called (by some authors) the dependence function. It follows that \(A(0)=A(1)=1\), and that \(A(\cdot)\) is a convex function with \(\max(x,1-x) \leq A(x)\leq 1\) for all \(0\leq x\leq1\). The lower and upper limits of \(A\) are obtained under complete dependence and independence respectively. \(A(\cdot)\) does not depend on the marginal parameters.
Some authors take B(x) = A(1-x) as the dependence function. If the
argument rev = TRUE
, then B(x) is plotted/evaluated.
abvnonpar
, fbvevd
,
rbvevd
, amvevd
abvevd(dep = 2.7, model = "hr")
abvevd(seq(0,1,0.25), dep = 0.3, asy = c(.7,.9), model = "alog")
abvevd(alpha = 0.3, beta = 1.2, model = "negbi", plot = TRUE)
bvdata <- rbvevd(100, dep = 0.7, model = "log")
M1 <- fitted(fbvevd(bvdata, model = "log"))
abvevd(dep = M1["dep"], model = "log", plot = TRUE)
abvnonpar(data = bvdata, add = TRUE, lty = 2)
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