ExtremalDep (version 1.0.0)

index.ExtDep: Index of extremal dependence

Description

Computes the extremal coefficient, Pickands dependence function, and the coefficients of upper and lower tail dependence for several parametric models. Also computes the lower tail dependence function for the bivariate skew-normal distribution.

Usage

index.ExtDep(object, model, par, x, u)

Value

• object="extremal": returns a value in \([1, d]\) (\(d=2,3\)).

• object="pickands": returns a value in \([\max(x), 1]\).

• object="upper.tail": returns a value in \([0, 1]\).

• object="lower.tail": returns a value in \([-1, 1]\).

Arguments

object

A character string indicating the index of extremal dependence to compute. Options are:

• "extremal": extremal coefficient,

• "pickands": Pickands dependence function,

• "upper.tail": coefficient of upper tail dependence,

• "lower.tail": coefficient of lower tail dependence.

model

A character string indicating the model/distribution.

• For object="extremal", "pickands", or "upper.tail": Husler-Reiss ("HR"), extremal-t ("ET"), extremal skew-t ("EST").

• For object="lower.tail": extremal-t ("ET"), extremal skew-t ("EST"), or skew-normal ("SN").

par

A vector of parameter values for the specified model/distribution.

x

A vector on the bivariate or trivariate unit simplex indicating where to evaluate the Pickands dependence function.

u

A real number in \([0,1]\) indicating the value at which to evaluate the lower tail dependence function of the bivariate skew-normal distribution.

Author

Simone Padoan https://faculty.unibocconi.it/simonepadoan/
Boris Beranger https://www.borisberanger.com

Details

The extremal coefficient is defined as $$\theta = V(1,\ldots,1) = d \int_{W} \max_{j \in \{1, ..., d\}} (w_j) dH(w) = - \log G(1,\ldots,1),$$ where \(W\) is the unit simplex, \(V\) is the exponent function, and \(G(\cdot)\) the distribution function of a multivariate extreme value model. Bivariate and trivariate versions are available.

The Pickands dependence function is defined as $$A(x) = - \log G(1/x)$$ for \(x\) in the bivariate/trivariate simplex \(W\).

The coefficient of upper tail dependence is defined as $$\vartheta = R(1,\ldots,1) = d \int_{W} \min_{j \in \{1, ..., d\}} (w_j) dH(w).$$ In the bivariate case, using the inclusion-exclusion principle this reduces to $$\vartheta = 2 + \log G(1,1) = 2 - V(1,1).$$

For the skew-normal distribution, the lower tail dependence function is defined as in Bortot (2010). This approximation is obtained in the limiting case as u tends to \(1\). The par argument should be a vector of length \(3\), consisting of the correlation parameter (between \(-1\) and \(1\)) and two real-valued skewness parameters.

References

Bortot, P. (2010). Tail dependence in bivariate skew-normal and skew-t distributions. Unpublished.

Examples

#############################
### Extremal skew-t model ###
#############################

## Extremal coefficient
index.ExtDep(object = "extremal", model = "EST", par = c(0.5, 1, -2, 2))

## Pickands dependence function
w <- seq(0.00001, 0.99999, length = 100)
pick <- numeric(100)
for (i in 1:100) {
  pick[i] <- index.ExtDep(
    object = "pickands", model = "EST", par = c(0.5, 1, -2, 2),
    x = c(w[i], 1 - w[i])
  )
}

plot(w, pick, type = "l", ylim = c(0.5, 1), ylab = "A(t)", xlab = "t")
polygon(c(0, 0.5, 1), c(1, 0.5, 1), lwd = 2, border = "grey")

## Upper tail dependence coefficient
index.ExtDep(object = "upper.tail", model = "EST", par = c(0.5, 1, -2, 2))

## Lower tail dependence coefficient
index.ExtDep(object = "lower.tail", model = "EST", par = c(0.5, 1, -2, 2))

################################
### Skew-normal distribution ###
################################

## Lower tail dependence function
index.ExtDep(object = "lower.tail", model = "SN", par = c(0.5, 1, -2), u = 0.5)