beed {ExtremalDep}  R Documentation 
Estimates the Pickands dependence function corresponding to multivariate data on the basis of a Bernstein polynomials approximation.
beed(data, x, d = 3, est = c("ht","cfg","md","pick"),
margin = c("emp","est","exp","frechet","gumbel"),
k = 13, y = NULL, beta = NULL, plot = FALSE)
data 

x 

d 
positive integer greater than or equal to two indicating the number of variables ( 
est 
string, indicating the estimation method ( 
margin 
string, denoting the type marginal distributions ( 
k 
postive integer, indicating the order of Bernstein
polynomials ( 
y 
numeric vector (of size 
beta 
vector of polynomial coefficients (see Details). 
plot 
logical; if 
The routine returns an estimate of the Pickands dependence function using the Bernstein polynomials approximation
proposed in Marcon et al. (2017).
The method is based on a preliminary empirical estimate of the Pickands dependence function.
If you do not provide such an estimate, this is computed by the routine. In this case, you can select one of the empirical methods
available. est = 'ht'
refers to the HallTajvidi estimator (Hall and Tajvidi 2000).
With est = 'cfg'
the method proposed by Caperaa et al. (1997) is considered. Note that in the multivariate case the adjusted version of Gudendorf and Segers (2011) is used. Finally, with est = 'md'
the estimate is based on the madogram defined in Marcon et al. (2017).
Each row of the (m \times d)
design matrix x
is a point in the unit d
dimensional simplex,
S_d := \left\{ (w_1,\ldots, w_d) \in [0,1]^{d}: \sum_{i=1}^{d} w_i = 1 \right\}.
With this "regularization"" method, the final estimate satisfies the neccessary conditions in order to be a Pickands dependence function.
A(\bold{w}) = \sum_{\bold{\alpha} \in \Gamma_k} \beta_{\bold{\alpha}} b_{\bold{\alpha}} (\bold{w};k).
The estimates are obtained by solving an optimization quadratic problem subject to the constraints. The latter are represented
by the following conditions:
A(e_i)=1; \max(w_i)\leq A(w) \leq 1; \forall i=1,\ldots,d;
(convexity).
The order of polynomial k
controls the smoothness of the estimate. The higher k
is, the smoother the final estimate is.
Higher values are better with strong dependence (e. g. k = 23
), whereas small values (e.g. k = 6
or k = 10
) are enough with mild or weak dependence.
An empirical transformation of the marginals is performed when margin="emp"
. A maxlikelihood fitting of the GEV distributions is implemented when margin="est"
. Otherwise it refers to marginal parametric GEV theorethical distributions (margin = "exp", "frechet", "gumbel"
).
beta 
vector of polynomial coefficients 
A 
numeric vector of the estimated Pickands dependence function 
Anonconvex 
preliminary nonconvex function 
extind 
extremal index 
The number of coefficients depends on both the order of polynomial k
and the dimension d
. The number of parameters is explained in Marcon et al. (2017).
The size of the vector beta
must be compatible with the polynomial order k
chosen.
With the estimated polynomial coefficients, the extremal coefficient, i.e. d*A(1/d,\ldots,1/d)
is computed.
Simone Padoan, simone.padoan@unibocconi.it, https://mypage.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com https://www.borisberanger.com/; Giulia Marcon, giuliamarcongm@gmail.com
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, 117.
x < ExtremalDep:::simplex(2)
data < evd::rbvevd(50, dep = 0.4, model = "log", mar1 = c(1,1,1))
Amd < beed(data, x, 2, "md", "emp", 20, plot=TRUE)
Acfg < beed(data, x, 2, "cfg", "emp", 20)
Aht < beed(data, x, 2, "ht", "emp", 20)
lines(x[,1], Aht$A, lty = 1, col = 3)
lines(x[,1], Acfg$A, lty = 1, col = 2)
##################################
# Trivariate case
##################################
if (interactive()){
x < ExtremalDep:::simplex(3)
data < evd::rmvevd(50, dep = 0.8, model = "log", d = 3, mar = c(1,1,1))
op < par(mfrow=c(1,3))
Amd < beed(data, x, 3, "md", "emp", 18, plot=TRUE)
Acfg < beed(data, x, 3, "cfg", "emp", 18, plot=TRUE)
Aht < beed(data, x, 3, "ht", "emp", 18, plot=TRUE)
par(op)
}