evd (version 2.2-1)

abvevd: Parametric Dependence Functions of Bivariate Extreme Value Models

Description

Calculate or plot the dependence function $A$ for nine parametric bivariate extreme value models.

Usage

abvevd(x = 0.5, dep, asy = c(1,1), alpha, beta, model = "log",
     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)", ...)

Arguments

x

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.

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

rev

Logical; reverse the dependence function? This is equivalent to evaluating the function at 1-x.

plot

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

add

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

lty, blty

Function and border line types. Set blty to zero to omit the border.

lwd, blwd

Function an border line widths.

col

Line colour.

xlim, ylim

x and y-axis limits.

xlab, ylab

x and y-axis labels.

...

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

Value

• abvevd calculates or plots the dependence function for one of nine parametric bivariate extreme value models, at specified parameter values.

synopsis

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)", ...)

Details

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.

See Also

abvnonpar, fbvevd, rbvevd, amvevd

Examples

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)