gen.ds {satdad}

Generate a Mevlog tail dependence structure.

Description

The function gen.ds creates (possibly randomly) a tail dependence structure for a multivariate extreme value logistic (Mevlog) model.

Usage

gen.ds(d, type = "alog", sub = NULL, dep = NULL, asy = NULL, mnns = d)

Arguments

d

The dimension.

type

The type of the model; represented by a character string. This is similar to the option model of rmvevd. It must be either "log" or "alog" (the default), for the symmetric logistic and the asymmetric logistic model respectively.

sub

An optional list of subsets of \{1,...,d\} involved in the tail dependence structure. If type = "log", then sub should be given by (1,\ldots,d), which is the way the code NULL will be interpreted. If type = "alog" and sub = NULL then a random list of vectors, subsets of \{1,...,d\}, is created. The cardinality of non singleton subsets in sub is given by mnns. If the user provides sub, it has to be a list of vectors, subsets of \{1,...,d\}, where each component from \{1,...,d\} appears at least once; Otherwise, one should add the missing singleton(s).

dep

An optional vector of dependence parameter(s). If type = "log", dep should be a single value. Otherwise, if type = "alog" and if the list sub is provided, then the length of the vector dep should be equal to that of the list sub (or a single value that will be replicated the length of sub times). Among these values, the dependence parameters associated singletons have to be equal to one. Otherwise, the values of dep associated to singleton will be ignored (and set to one). When dep = NULL its values are randomly generated.

asy

An optional list of asymmetric weights. If type = "log", then asy should be the vector (1,\ldots,1), which is the way the code NULL will be interpreted. If type = "alog" and if sub is provided, the length of the list asy should be in accordance with the length of sub. If asy = NULL then the values are randomly generated. Note that asy satisfies the sum-to-one constraints.

mnns

The default value is arbitrarily equal to d. When sub = NULL, the list sub is randomly generated, and its size is closely related to mnns. The latter represents the number of non singletons subsets included in sub.

Details

A multivariate extreme value logistic (Mevlog) model is symmetric or asymmetric.

• type = "log". It generates a multivariate symmetric logistic model. Such model is a well-known generalization of the bivariate extreme value logistic model introduced by Gumbel (1960). The parameter 'dep' (with 0 < `dep` \leq 1) is the only parameter needed to write the following equation

  \ell(u) = ( \sum_{i=1}^d u_i^{1/\code{dep}} )^{\code{dep}}.

  If the parameter dep is missing, the function gen.ds will randomly generate its value from a standard uniform distribution. The list asy is reduced to a vector of ones whereas the list sub only contains the maximal vector (1, \ldots, d).

  This is a special case of the multivariate asymmetric logistic model (alog case).

• type = "alog". It generates a multivariate asymmetric logistic model, which has been first introduced by Tawn (1990). We have

  \ell(u)=\sum_{b\in B} (\sum_{i \in b} (\beta_{i,b}u_i)^{1/\alpha_b})^{\alpha_b}

  where B is the power set of \{1,...,d\} (or a strict subset of the power set), the dependence parameters \alpha_b lie in (0,1] and the collection of asymmetric weights \beta_{i,b} are coefficients from [0,1] satisfying \forall i \in \{1,\ldots,d\}, \sum_{b\in B: i \in b} \beta_{i,b}=1. Missing asymmetric weights \beta_{i,b} are assumed to be zero.

The function gen.ds generates here an object of class ds which corresponds in this package to the stable tail dependence function \ell. The class ds consists of:

• the dimension d.

• the type "log" or alog.

• the list sub that corresponds to B. When sub is provided, the same list of subsets is returned, eventually sorted. When sub = NULL then sub is a list of subsets of the power set of \{1,...,d\}. When the option mnns is used, the latter integer indicates the cardinality of non singleton subsets in B.

  the dependence parameter dep or the vector of dependence parameters dep. When missing, these coefficients are obtained from independent standard uniform sampling.

  the list asy of asymmetric weights \beta_{i,b} for b \in B and i \in b. When missing, these coefficients are obtained from independent standard uniform sampling followed by renormalization in order to satisfy the sum-to-one constraints.

Value

gen.ds returns an object representing a tail dependence structure for Mevlog models. Such object is a list containing the following components:

• d The dimension.

• type The type of the model either "log" or "alog".

• sub The list of subsets of \{1,...,d\} involved in the tail dependence support.

• dep The vector of dependence parameter(s).

• asy The list of asymmetric weights.

Note

The first interest of the gen.ds function is to generate randomly a tail dependence structure. Since sub and asy become quickly very large lists as d increases, it is very convenient to obtain automatically well-defined tail dependence structures for multivariate extreme value logistic models.

The second interest of the gen.ds function is to produce partial models where all subsets do not necessarily contribute to the tail dependence support.

The function gen.ds does not manage margins characteristics which will be handle by the option mar in the r-d-p-Mevlog functions.

Author(s)

Cécile Mercadier ([e-mail])

References

Gumbel, E. J. (1960) Distributions des valeurs extremes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris, 9, 171–173.

Stephenson, A. (2002) evd: Extreme Value Distributions. R News, 2(2):31–32.

Tawn, J. A. (1990) Modelling multivariate extreme value distributions. Biometrika, 77, 245–253.

See Also

ellMevlog, graphs

Examples

## Fix a 5-dimensional symmetric tail dependence structure
## The dependence paramater is fixed to .7
(ds5 <- gen.ds(d = 5, dep = .7, type = "log"))

## Fix a 3-dimensional asymmetric tail dependence structure
## The list sub and asy are provided ; The vector dep is randomly generated
(ds3 <- gen.ds(d = 3, sub = list(c(1,2), c(1,2,3)), asy = list(c(0.4,0.6), c(0.6,0.4,1))))
graphs(ds = ds3)

## Fix a 8-dimensional asymmetric tail dependence structure
## The lists sub and asy, as the vector dep, are randomly generated
(ds8 <- gen.ds(d = 8))
graphs(ds = ds8)