gethac: gethac

In VecDep: Measuring Copula-Based Dependence Between Random Vectors

gethac

Description

Given a q-dimensional random vector \mathbf{X} = (\mathbf{X}_{1},...,\mathbf{X}_{k}) with \mathbf{X}_{i} a d_{i}-dimensional random vector, i.e., q = d_{1} + ... + d_{k}, this function construct a hac object (hierarchical Archimedean copula) with two nesting levels given the specified dimensions and parameters of the root and k child copulas.

Usage

gethac(dim, thetas, type)

Arguments

dim	The vector of dimensions (d_{1},...,d_{k}).
thetas	The parameters (\theta_{0}, \theta_{1}, \dots, \theta_{k}).
type	The type of Archimedean copula.

Details

A hierarchical (or nested) Archimedean copula C with two nesting levels and k child copulas is given by

C(\mathbf{u}) = C_{0} \left (C_{1}(\mathbf{u}_{1}), \dots, C_{k}(\mathbf{u}_{k}) \right ),

where \mathbf{u} = (\mathbf{u}_{1}, \dots, \mathbf{u}_{k}) \in \mathbb{R}^{q} with \mathbf{u}_{i} \in \mathbb{R}^{d_{i}} for i = 1, \dots, k. The (k-dimensional) copula C_{0} is called the root copula, and the (d_{i}-dimensional) copulas C_{i} are the child copulas.

They all belong to the class of Archimedean copulas, and we denote \theta_{i} for the parameter of C_{i} for i = 0,1,\dots,k. A sufficient condition to guarantee that C indeed is a copula, is that C_{0},C_{1}, \dots, C_{k} are all a particular member of this class of Archimedean copulas (e.g., Clayton), and such that \theta_{0} \leq \theta_{i} for i = 1, \dots, k (sufficient nesting condition).

When a certain child copula C_{i} is one dimensional (\mathbf{X}_{i} is one dimensional), \theta_{i} can be any number. It must hold that length(thetas) = k + 1.

Many functions for working with nested Archimedean copulas are developed in the R package ‘HAC’, and the function gethac utilizes these functions to quickly construct a hac object that is useful for modelling the dependence between \mathbf{X}_{1}, \dots, \mathbf{X}_{k}. See also the R package ‘HAC’ for the different possibilities of type (specified by a number in \{1,\dots,10\}).

Value

A hac object with two nesting levels and k child copulas.

References

De Keyser, S. & Gijbels, I. (2024). Parametric dependence between random vectors via copula-based divergence measures. Journal of Multivariate Analysis 203:105336.
doi: https://doi.org/10.1016/j.jmva.2024.105336.

Okhrin, O., Ristig, A. & Chen, G. (2024). HAC: estimation, simulation and visualization of hierarchical Archimedean copulae (HAC), R package version 1.1-1.
url: https://CRAN.R-project.org/package=HAC.

See Also

phihac for computing the \Phi-dependence between all the child copulas of a hac object with two nesting levels, Helhac for computing the Hellinger distance between all the child copulas of a hac object with two nesting levels, mlehac for maximum pseudo-likelihood estimation of the parameters of a hac object with two nesting levels.

Examples

dim = c(3,5,1,2)
thetas = c(2,2,3,1,4)

# 11 dimensional nested Gumbel copula with
# (theta_0,theta_1,theta_2,theta_3,theta_4) = (2,2,3,1,4),
# where the value of theta_3 could be anything,
# because the third random vector is one dimensional

HAC = gethac(dim,thetas,type = 1)

VecDep documentation built on April 4, 2025, 5:14 a.m.