Calculate or plot the density \(h\) of the spectral measure \(H\) on the interval \((0,1)\), for nine parametric bivariate extreme value models.
hbvevd(x = 0.5, dep, asy = c(1,1), alpha, beta, model = c("log", "alog",
"hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"),
half = FALSE, plot = FALSE, add = FALSE, lty = 1, ...)
hbvevd
calculates or plots the spectral density function
\(h\) for one of nine parametric bivariate extreme value models, at specified parameter values.
A vector of values at which the function is evaluated
(ignored if plot or add is TRUE
). \(h(1/2)\) is
returned by default.
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; if TRUE
the function is divided by two,
corresponding to a spectral measure with total mass one rather
than two.
Logical; if TRUE
the function is plotted. The
x and y values used to create the plot are returned invisibly.
Logical; add to an existing plot?
Line type.
Other high-level graphics parameters to be passed to
plot
.
For differentiable models H may have up to two point masses:
at zero and one. Assuming that the model parameters are in the
interior of the parameter space, we have the following. For the
asymmetric logistic and asymmetric negative logistic models the
point masses are of size 1-asy1
and 1-asy2
respectively. For the asymmetric mixed model they are of size
1-alpha-beta
and 1-alpha-2*beta
respectively. For
all other models the point masses are zero.
At independence, H has point masses of size one at both zero and one. At complete dependence [a non-differentiable model] H has a single point mass of size two at \(1/2\). In either case, \(h\) is zero everywhere.
Any bivariate extreme value distribution can be written as
$$G(z_1,z_2) = \exp\left[-\int_0^1 \max\{wy_1, (1-w)y_2\}
H(dw)\right]$$
for some function \(H(\cdot)\) defined on \([0,1]\),
satisfying
$$\int_0^1 w H(dw) = \int_0^1 (1-w) H(dw) = 1$$
In particular, the total mass of H is two.
The functions \(y_1\) and \(y_2\) are as defined in
abvevd
.
H is called the spectral measure, with density \(h\) on the interval \((0,1)\).
abvevd
, fbvevd
,
rbvevd
, plot.bvevd
hbvevd(dep = 2.7, model = "hr")
hbvevd(seq(0.25,0.5,0.75), dep = 0.3, asy = c(.7,.9), model = "alog")
hbvevd(alpha = 0.3, beta = 1.2, model = "negbi", plot = TRUE)
bvdata <- rbvevd(100, dep = 0.7, model = "log")
M1 <- fitted(fbvevd(bvdata, model = "log"))
hbvevd(dep = M1["dep"], model = "log", plot = TRUE)
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