dens {ExtremalDep} | R Documentation |

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.

```
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)
```

`x` |
A ( |

`model` |
A string with the name of the parametric model to be estimated. Models are
Pairwise Beta ( |

`par` |
A vector containing the parameters of the model. See |

`c` |
A real value in |

`log` |
Logical; if |

`vectorial` |
Logical; if |

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,

`\mu \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,

`\mu \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.

Returns a `n`

-dimensional vector if `vectorial=TRUE`

or a single value
if `vectorial=FALSE`

.

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

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.

```
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)
}
```

[Package *ExtremalDep* version 0.0.3-5 Index]