evd (version 2.2-1)

hbvevd: Parametric Spectral Density Functions of Bivariate Extreme Value Models

Description

Calculate or plot the density $h$ of the spectral measure $H$ on the interval $(0,1)$, for nine parametric bivariate extreme value models.

Usage

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, ...)

Arguments

x

A vector of values at which the function is evaluated (ignored if plot or add is TRUE). $h(1/2)$ is returned by default.

dep

Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.

asy

A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.

alpha, beta

Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.

model

The specified model; a character string. Must be either "log" (the default), "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct" o

half

Logical; if TRUE the function is divided by two, corresponding to a spectral measure with total mass one rather than two.

plot

Logical; if TRUE the function is plotted. The x and y values used to create the plot are returned invisibly.

add

Logical; add to an existing plot?

lty

Line type.

...

Other high-level graphics parameters to be passed to plot.

Value

• hbvevd calculates or plots the spectral density function $h$ for one of nine parametric bivariate extreme value models, at specified parameter values.

Point Masses

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.

Details

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.

$H(\cdot)$ is called the spectral measure, with density $h$ on the interval $(0,1)$.

See Also

abvevd, fbvevd, rbvevd, plot.bvevd

Examples

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)