Computes a non-parametric estimate of the Pickands dependence function \(A(w)\) for multivariate data, based on the madogram estimator.
madogram(w, data, margin = c("emp", "est", "exp", "frechet", "gumbel"))A numeric vector of estimates of the Pickands dependence function.
An \(m \times d\) design matrix (see Details).
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.
A string indicating the type of marginal distributions
("emp" by default, see Details).
Simone Padoan, simone.padoan@unibocconi.it, https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com, https://www.borisberanger.com; Giulia Marcon, giuliamarcongm@gmail.com
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.
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.
beed, beed.confband
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)
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