POT (version 1.1-11)

pickdep: The Pickands' Dependence Function

Description

Return and optionally plot the Pickands' dependence function.

Usage

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

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.

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.

Author

Mathieu Ribatet

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, \(A(w) = \max (w, 1-w)\).

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)