Computes a non-parametric estimate 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.
\((m \times d)\) design matrix (see Details).
\((n \times d)\) matrix of data or data frame with d columns. d is the numer of variables and n is the number of replications.
string, denoting the type marginal distributions (margin="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
\( \nu(\bold{w}) = {\rm E} \left(\ \bigvee_{i=1,\dots,d}\left \lbrace F^{1/w_i}_{i}\left(X_{i}\right) \right\rbrace - \frac{1}{d}\sum_{i=1,\dots,d}F^{1/w_i}_{i}\left(X_{i}\right). \right), \)
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 unit
d-dimensional simplex.
If \(X\) is a d-dimensional max-stable distributed random vector, with exponent measure function \(V(\bold{x})\) and Pickands dependence function \(A(\bold{w})\), then
\(\nu(\bold{w})=V(1/w_1,\ldots,1/w_d)/(1+V(1/w_1,\ldots,1/w_d))-c(\bold{w}),\) where \(c(\bold{w})=d^{-1}\sum_{i=1}^{d}{w_i/(1+w_i)}\).
From this, it follows that
\( V(1/w_1,\ldots,1/w_d)=\frac{\nu(\bold{w})+c(\bold{w})}{1-\nu(\bold{w})-c(\bold{w})}, \)
and
\( A(\bold{w})=\frac{\nu(\bold{w})+c(\bold{w})}{1-\nu(\bold{w})-c(\bold{w})}. \)
An empirical transformation of the marginals is performed when margin="emp".
A max-likelihood fitting of the GEV distributions is implemented when margin="est".
Otherwise it refers to marginal parametric GEV theorethical distributions (margin="exp", "frechet", "gumbel").
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., 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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