pickdep {POT}

The Pickands' Dependence Function

Description

Return and optionally plot the Pickands' dependence function.

Usage

pickdep(object, main, bound = TRUE, plot = TRUE, ...)

Arguments

object

A object of class bvpot. Usually, object is the return of function fitbvgpd.

main

May be missing. If present, the plot title.

bound

Logical. Should the perfect dependent and independent case bounds be plotted?

plot

Logical. Should the dependence function be plotted?

...

Optional parameters to be passed to the lines function.

Details

It is common to parametrize a bivariate extreme value distribution according to the Pickands' representation (Pickands, 1981). That is, if G is any bivariate extreme value distribution, then it has the following parametrization:

G\left(y_1,y_2\right) = \exp\left[- \left(\frac{1}{z_1} + \frac{1}{z_2} \right) A\left( \frac{z_2}{z_1+z_2} \right) \right]

where z_i are unit Frechet.

A is the Pickands' dependence function. It has the following properties:

• A is defined on [0,1];

• A(0)=A(1)=1;

• \max \left(w, 1-w \right) \leq A(w) \leq 1, \quad \forall w;

• A is a convex function;

• For two independent (unit Frechet) random variables, A(w) = 1, \quad \forall w;

• For two perfectly dependent (unit Frechet) random variables, A(w) = \max (w, 1-w).

Value

The function returns an invisible function: the Pickands' dependence function. Moreover, the returned object has an attribute which specifies the model for the bivariate extreme value distribution.

If plot = TRUE, then the dependence function is plotted.

Author(s)

Mathieu Ribatet

References

Pickands, J. (1981) Multivariate Extreme Value Distributions Proceedings 43rd Session International Statistical Institute

Examples

x <- rbvgpd(1000, alpha = 0.9, model = "mix", mar1 = c(0,1,0.25),
 mar2 = c(2,0.5,0.1))
Mmix <- fitbvgpd(x, c(0,2), "mix")
pickdep(Mmix)