## Madogram-based estimation of the Pickands Dependence Function

### Description

Computes a non-parametric estimate 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 (m \times d) design matrix (see Details). data (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. margin string, denoting the type marginal distributions (margin="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(\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").

### Value

A numeric vector of estimates.

### 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., Diebolt, J. (2009) Modelling pairwise dependence of maxima in space, Biometrika, 96(1), 1-17.

beed, beed.confband

### Examples

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