dExtDep {ExtremalDep}

Parametric and non-parametric density of Extremal Dependence

Description

This function calculates the density of parametric multivariate extreme distributions and corresponding angular density, or the non-parametric angular density represented through Bernstein polynomials.

Usage

dExtDep(x, method="Parametric", model, par, angular=TRUE, log=FALSE, 
        c=NULL, vectorial=TRUE, mixture=FALSE)

Arguments

x

A vector or a matrix. The value at which the density is evaluated.

method

A character string taking value "Parametric" or "NonParametric"

model

A string with the name of the model: "PB" (Pairwise Beta), "HR" (Husler-Reiss), "ET" (Extremal-t), "EST" (Extremal Skew-t), TD (Tilted Dirichlet) or AL (Asymmetric Logistic) when evaluating the angular density. Restricted to "HR", "ET" and "EST" for multivariate extreme value densities. Required when method="Parametric".

par

A vector representing the parameters of the (parametric or non-parametric) model.

angular

A logical value specifying if the angular density is computed.

log

A logical value specifying if the log density is computed.

c

A real value in [0,1], providing the decision rule to allocate a data point to a subset of the simplex. Only required for the parametric angular density of the Extremal-t, Extremal Skew-t and Asymmetric Logistic models.

vectorial

A logical value; if TRUE a vector of nrow(x) densities is returned. If FALSE the likelihood function is returned (product of densities or sum if log=TRUE. TRUE is the default.

mixture

A logical value specifying if the Bernstein polynomial representation of distribution should be expressed as a mixture. If mixture=TRUE the density integrates to 1. Required when method="NonParametric".

Details

Note that when method="Parametric" and angular=FALSE, the density is only available in 2 dimensions. When method="Parametric" and angular=TRUE, the models "AL", "ET" and "EST" are limited to 3 dimensions. This is because of the existence of mass on the subspaces on the simplex (and therefore the need to specify c).

For the parametric models, the appropriate length of the parameter vector can be obtained from the dim_ExtDep function and are summarized as follows. When model="HR", the parameter vector is of length choose(dim,2). When model="PB" or model="Extremalt", the parameter vector is of length choose(dim,2) + 1. When model="EST", the parameter vector is of length choose(dim,2) + dim + 1. When model="TD", the parameter vector is of length dim. When model="AL", the parameter vector is of length 2^(dim-1)*(dim+2) - (2*dim+1).

Value

If x is a matrix and vectorial=TRUE, a vector of length nrow(x), otherwise a scalar.

Author(s)

Simone Padoan, [e-mail], https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, [e-mail] https://www.borisberanger.com;

References

Beranger, B. and Padoan, S. A. (2015). Extreme dependence models, chapater of the book Extreme Value Modeling and Risk Analysis: Methods and Applications, Chapman Hall/CRC.

Beranger, B., Padoan, S. A. and Sisson, S. A. (2017). Models for extremal dependence derived from skew-symmetric families. Scandinavian Journal of Statistics, 44(1), 21-45.

Coles, S. G., and Tawn, J. A. (1991), Modelling Extreme Multivariate Events, Journal of the Royal Statistical Society, Series B (Methodological), 53, 377–392.

Cooley, D.,Davis, R. A., and Naveau, P. (2010). The pairwise beta distribution: a flexible parametric multivariate model for extremes. Journal of Multivariate Analysis, 101, 2103–2117.

Engelke, S., Malinowski, A., Kabluchko, Z., and Schlather, M. (2015), Estimation of Husler-Reiss distributions and Brown-Resnick processes, Journal of the Royal Statistical Society, Series B (Methodological), 77, 239–265.

Husler, J. and Reiss, R.-D. (1989), Maxima of normal random vectors: between independence and complete dependence, Statistics and Probability Letters, 7, 283–286.

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.

Nikoloulopoulos, A. K., Joe, H., and Li, H. (2009) Extreme value properties of t copulas. Extremes, 12, 129–148.

Opitz, T. (2013) Extremal t processes: Elliptical domain of attraction and a spectral representation. Jounal of Multivariate Analysis, 122, 409–413.

Tawn, J. A. (1990), Modelling Multivariate Extreme Value Distributions, Biometrika, 77, 245–253.

See Also

pExtDep, rExtDep, fExtDep, fExtDep.np

Examples

# Example of PB on the 4-dimensional simplex
dExtDep(x=rbind(c(0.1,0.3,0.3,0.3),c(0.1,0.2,0.3,0.4)), method="Parametric", 
        model="PB", par=c(2,2,2,1,0.5,3,4), log=FALSE)

# Example of EST in 2 dimensions
dExtDep(x=c(1.2,2.3), method="Parametric", model="EST", par=c(0.6,1,2,3), angular=FALSE, log=TRUE)

# Example of non-parametric angular density
beta <- c(1.0000000, 0.8714286, 0.7671560, 0.7569398, 
          0.7771908, 0.8031573, 0.8857143, 1.0000000)
dExtDep(x=rbind(c(0.1,0.9),c(0.2,0.8)), method="NonParametric", par=beta)