evd (version 2.0-1)

abvpar: Parametric Dependence Functions of Bivariate Extreme Value Models

Description

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

Usage

abvpar(x = 0.5, dep, asy = c(1,1), alpha, beta, model = "log",
     plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3,
     xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...)

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 and Coles-Tawn models.

model

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

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 abvpar 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

Line width.

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

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

synopsis

abvpar(x = 0.5, dep, asy = c(1,1), alpha, beta, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct"), plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1, blty = 3, xlim = c(0,1), ylim = c(0.5,1), xlab = "", ylab = "", ...)

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.

See Also

abvnonpar, fbvevd, rbvevd, atvpar

Examples

abvpar(dep = 2.7, model = "hr")
abvpar(seq(0,1,0.25), dep = 0.3, asy = c(.7,.9), model = "alog")
abvpar(alpha = 0.3, beta = 1.2, model = "negbi", plot = TRUE)

bvdata <- rbvevd(100, dep = 0.7, model = "log")
M1 <- fitted(fbvevd(bvdata, model = "log"))
abvpar(dep = M1["dep"], model = "log", plot = TRUE)
abvnonpar(data = bvdata, add = TRUE, lty = 2)