dens: Angular density and likelihood function for some extremal dependence models

Description

Evaluates the angular density or calculates the likelihood function of the Pairwise Beta, Husler-Reiss, Dirichlet, Extremal-$t$, Extremal Skew-$t$ and Asymmetric Logistic models at one or more locations on the unit simplex.

Usage

dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Pairwise", par=c(2,2,2,4), c,
        log=FALSE, vectorial=TRUE)

Value

Returns a $n$ -dimensional vector if vectorial=TRUE or a single value if vectorial=FALSE.

Arguments

x: A ( $n \times d$ ) matrix of angular components, where the rows represent $n$ independent points in the $d$ -dimensional unit simplex. See Details. The default is rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), two points in the $3$ -dimensional simplex.
model: A string with the name of the parametric model to be estimated. Models are Pairwise Beta ("Pairwise"), Husler-Reiss ("Husler"), Dirichlet ("Dirichlet"), Extremal-t ("Extremalt"), Extremal Skew-t ("Skewt") and Asymmetric Logistic ("Asymmetric").
par: A vector containing the parameters of the model. See Details.
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 Extremal-t, Extremal Skew-t and Asymmetric Logistic models.
log: Logical; if TRUE the log-density is returned. FALSE is the default.
vectorial: Logical; if TRUE when $n > 1$ the different densities are returned as a vector of length $n$ . If FALSE the likelihood function is returned. TRUE is the default.

Author

Simone Padoan, simone.padoan@unibocconi.it, https://mypage.unibocconi.it/simonepadoan/; Boris Beranger, borisberanger@gmail.com https://www.borisberanger.com/;

Details

The Extremal-$t$ and Asymmetric Logistic models are available up to 3 dimensions; mass on all the subsets of the simplex is included.

For the Pairwise Beta model, the parameter vector is decomposed as:

b: A vector of size choose(d,2). Controls the dependence between pairs. The default is b=c(2,2,2).
alpha: A positive real that controls the general dependence between all the variables. The default is $4$ .

For the Husler-Reiss model, the parameter vector is of size choose(d,2).

For the Dirichlet model, the parameter vector is decomposed a vector of size $d$ which controls the dependence between pairs.

For the Extremal-$t$ model, the parameter vector is decomposed as:

rho: A vector of size choose(d,2) representing the corrleation parameters.
mu: A positive integer, $μ \geq 1$ , representing the degree of freedom.

For the Extremal Skew-$t$ model, the parameter vector is decomposed as:

rho: A vector of size choose(d,2) representing the corrleation parameters.
alpha: A vector of size d representing the shape parameters.
mu: A positive integer, $μ \geq 1$ , representing the degree of freedom.

For the Asymmetric Logistic model, the parameter vector is decomposed as:

alpha: A vector of size $1$ or $4$ depending on whether $d = 2$ or $3$ .
beta: A vector of size $2$ or $9$ depending on whether $d = 2$ or $3$ .

If log=TRUE and vectorial=FALSE then the log-likelihood function is calculated.

References

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.

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

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.

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

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.

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.

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

Examples

Run this code

if (interactive()){
### Pairwise Beta :


# Examples on the 3-dimensional simplex
# Returns the bivariate angular density at two locations

dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Pairwise", par=c(2,2,2,4),
log=FALSE, vectorial=TRUE)

# returns the likelihood function at two locations

dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Pairwise", par=c(2,2,2,4),
log=FALSE, vectorial=FALSE)

# returns the log-likelihood function

dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Pairwise", par=c(2,2,2,4),
log=TRUE, vectorial=FALSE)

# Examples on the 4-dimensional simplex
# returns the bivariate angular density at two locations

dens(x=rbind(c(0.1,0.3,0.3,0.3),c(0.1,0.2,0.3,0.4)), model="Pairwise", par=c(2,2,2,1,0.5,3,4),
log=FALSE, vectorial=TRUE)

# returns the likelihood function at two locations
dens(x=rbind(c(0.1,0.3,0.3,0.3),c(0.1,0.2,0.3,0.4)), model="Pairwise", par=c(2,2,2,1,0.5,3,4),
log=FALSE, vectorial=FALSE)

# returns the log-likelihood function

dens(x=rbind(c(0.1,0.3,0.3,0.3),c(0.1,0.2,0.3,0.4)), model="Pairwise", par=c(2,2,2,1,0.5,3,4),
log=TRUE, vectorial=FALSE)


### Husler-Reiss


# Example on the 2-dimensional simplex
# returns the log-likelihood at two locations

dens(x=rbind(c(0.1,0.9),c(0.3,0.7)), model="Husler", par=1.7,
log=TRUE, vectorial=FALSE)

# Example on the 3-dimensional simplex
# returns the likelihood function at two locations

dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Husler", par=c(1.7,0.7,1.1),
log=FALSE, vectorial=FALSE)

# Example on the 4-dimensional simplex
# returns the bivariate angular density at two locations

dens(x=rbind(c(0.1,0.1,0.4,0.4),c(0.1,0.2,0.3,0.4)), model="Husler", par=rep(1,6),
log=FALSE, vectorial=TRUE)


### Dirichlet


# Example on the 2-dimensional simplex
# returns the log-likelihood at two points

dens(x=rbind(c(0.1,0.9),c(0.3,0.7)), model="Dirichlet", par=c(1.7,0.7),
log=TRUE, vectorial=FALSE)


# Example on the 3-dimensional simplex
# returns the likelihood function at three locations

dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Dirichlet", par=c(1.7,0.7,1.1),
log=FALSE, vectorial=FALSE)

# Example on the 4-dimensional simplex
# returns the bivariate angular density at two locations

dens(x=rbind(c(0.1,0.1,0.4,0.4),c(0.1,0.2,0.3,0.4)), model="Dirichlet", par=c(1.7,0.7,1.1,0.1),
log=FALSE, vectorial=TRUE)


### Extremal-t


# Example on the 2-dimensional simplex
# Returns the log-likelihood

dens(x=rbind(c(0.4,0.6),c(0.3,0.7)), model="Extremalt", par=c(0.7,2), c=0.1,
log=TRUE, vectorial=FALSE)

# Density in the corner

dens(x=c(0.08,0.92), model="Extremalt", par=c(0.7,2), c=0.1,
log=FALSE, vectorial=FALSE)


# Example on the 3-dimensional simplex
# Returns the log-likelihood

dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Extremalt", par=c(rep(0.1,3),2), c=0.03,
log=FALSE, vectorial=FALSE)

# Returns the evalutaion of the angular density at three locations:
# The first one is set to be on the edge linking the second and third components
# The second one is set to be on the interior of the simplex
# The third one is set to be on the corner near the third component

if (interactive()){
dens(x=rbind(c(0.001,0.3,0.699),c(0.1,0.2,0.7),c(0.001,0.001,0.998)),
model="Extremalt", par=c(rep(0.1,3),2), c=0.01, log=FALSE, vectorial=TRUE)
}

### Extremal Skew-t


# Example on the 2-dimensional simplex
# Returns the log-likelihood

dens(x=rbind(c(0.4,0.6),c(0.3,0.7)), model="Skewt", par=c(0.7,0,0,2), c=0.1,
log=TRUE, vectorial=FALSE)

dens(x=rbind(c(0.4,0.6),c(0.3,0.7)), model="Skewt", par=c(0.7,2,-1,2), c=0.1,
log=TRUE, vectorial=FALSE)

# Density in the corner

dens(x=c(0.08,0.92), model="Skewt", par=c(0.7,0,0,2), c=0.1,
log=FALSE, vectorial=FALSE)

dens(x=c(0.08,0.92), model="Skewt", par=c(0.7,-1,2,2), c=0.1,
log=FALSE, vectorial=FALSE)

# Example on the 3-dimensional simplex
# Returns the log-likelihood

dens(x=rbind(c(0.1,0.3,0.6),c(0.1,0.2,0.7)), model="Skewt", par=c(rep(0.1,3),rep(0,3),2), c=0.03,
log=FALSE, vectorial=FALSE)

# Returns the evalutaion of the angular density at three locations:
# The first one is set to be on the edge linking the second and third components
# The second one is set to be on the interior of the simplex
# The third one is set to be on the corner near the third component

if (interactive()){
dens(x=rbind(c(0.001,0.3,0.699),c(0.1,0.2,0.7),c(0.001,0.001,0.998)),
model="Skewt", par=c(rep(0.1,3),rep(0,3),2), c=0.01, log=FALSE, vectorial=TRUE)
}


### Asymmetric Logistic

# Example on the 3-dimensional simplex
# Returns the angular density at three points:
# The first one is set to be on the edge linking the second and third components
# The second one is set to be on the interior of the simplex
# The third one is set to be on the corner near the third component

dens(x=rbind(c(0.001,0.3,0.699),c(0.1,0.2,0.7),c(0.001,0.001,0.998)), c=0.05,
model="Asymmetric", par=c(1.2,1.8,4,2,rep(0.3,9)), log=FALSE, vectorial=TRUE)


}

Run the code above in your browser using DataLab

Get 50% off unlimited learning