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").