madogram: Madogram-based estimation of the Pickands Dependence Function

In ExtremalDep: Extremal Dependence Models

Madogram-based estimation of the Pickands Dependence Function

Description

Computes a non-parametric estimate of the Pickands dependence function A(w) for multivariate data, based on the madogram estimator.

Usage

madogram(w, data, margin = c("emp", "est", "exp", "frechet", "gumbel"))

Arguments

w	An m \times d design matrix (see Details).
data	An n \times d matrix of data or data frame with d columns. Here, d is the number of variables and n the number of replications.
margin	A string indicating the type of marginal distributions ("emp" by default, see Details).

Details

The estimation procedure is based on the madogram as proposed in Marcon et al. (2017). The madogram is defined by

\nu(\mathbf{w}) = \mathbb{E}\left( \max_{i=1,\dots,d}\left\lbrace F_i^{1/w_i}(X_i) \right\rbrace - \frac{1}{d}\sum_{i=1}^d F_i^{1/w_i}(X_i) \right),

where 0 < w_i < 1 and w_d = 1 - (w_1 + \ldots + w_{d-1}).

Each row of the design matrix w is a point in the d-dimensional unit simplex.

If X is a d-dimensional max-stable random vector with exponent measure V(\mathbf{x}) and Pickands dependence function A(\mathbf{w}), then

\nu(\mathbf{w}) = \frac{V(1/w_1,\ldots,1/w_d)}{1 + V(1/w_1,\ldots,1/w_d)} - c(\mathbf{w}),

where

c(\mathbf{w}) = \frac{1}{d}\sum_{i=1}^d \frac{w_i}{1+w_i}.

From this, it follows that

V(1/w_1,\ldots,1/w_d) = \frac{\nu(\mathbf{w}) + c(\mathbf{w})}{1 - \nu(\mathbf{w}) - c(\mathbf{w})},

and

A(\mathbf{w}) = \frac{\nu(\mathbf{w}) + c(\mathbf{w})}{1 - \nu(\mathbf{w}) - c(\mathbf{w})}.

Marginal treatment:

• "emp": empirical transformation of the marginals,

• "est": maximum-likelihood fitting of the GEV distributions,

• "exp", "frechet", "gumbel": parametric GEV theoretical distributions.

Value

A numeric vector of estimates of the Pickands dependence function.

Author(s)

Simone Padoan, [e-mail], https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, [e-mail], https://www.borisberanger.com; Giulia Marcon, [e-mail]

References

Marcon, G., Padoan, S.A., Naveau, P., Muliere, P. and Segers, J. (2017). Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials. Journal of Statistical Planning and Inference, 183, 1–17.

Naveau, P., Guillou, A., Cooley, D. and Diebolt, J. (2009). Modelling pairwise dependence of maxima in space. Biometrika, 96(1), 1–17.

See Also

beed, beed.confband

Examples

x <- simplex(2)
data <- evd::rbvevd(50, dep = 0.4, model = "log", mar1 = c(1,1,1))

Amd <- madogram(x, data, "emp")
Amd.bp <- beed(data, x, 2, "md", "emp", 20, plot = TRUE)

lines(x[,1], Amd, lty = 1, col = 2)

ExtremalDep documentation built on Aug. 21, 2025, 5:57 p.m.