htrafo: GOF Testing Transformation of Hering and Hofert

Description

The transformation described in Hering and Hofert (2014), for Archimedean copulas.

Usage

htrafo(u, copula, include.K = TRUE, n.MC = 0, inverse = FALSE,
       method = eval(formals(qK)$method), u.grid, ...)

Value

htrafo() returns an

$n\times d$- or $n\times (d-1)$-matrix (depending on whether include.K is TRUE or

FALSE) containing the transformed input u.

Arguments

u: $n\times d$-matrix with values in $[0,1]$. If inverse=FALSE (the default), u contains (pseudo-/copula-)observations from the copula copula based on which the transformation is carried out; consider applying the function pobs() first in order to obtain u. If inverse=TRUE, u contains $U[0,1]^d$ distributed values which are transformed to copula-based (copula) ones.
copula: an Archimedean copula specified as "outer_nacopula" or "archmCopula".
include.K: logical indicating whether the last component, involving the Kendall distribution function K, is used in htrafo().
n.MC: parameter n.MC for K.
inverse: logical indicating whether the inverse of the transformation is returned.
method: method to compute qK().
u.grid: argument of qK() (for method="discrete").
...: additional arguments passed to qK() if inverse = TRUE.

Details

Given a $d$-dimensional random vector $\bm{U}$ following an Archimedean copula $C$ with generator $\psi$, Hering and Hofert (2014) showed that $\bm{U}^\prime\sim\mathrm{U}[0,1]^d$, where $$U_{j}^\prime=\left(\frac{\sum_{k=1}^{j}\psi^{-1}(U_{k})}{ \sum_{k=1}^{j+1}\psi^{-1}(U_{k})}\right)^{j},\ j\in\{1,\dots,d-1\},\ U_{d}^\prime=K(C(\bm{U})).$$ htrafo applies this transformation row-wise to u and thus returns either an $n\times d$- or an $n\times (d-1)$-matrix, depending on whether the last component $U^\prime_d$ which involves the (possibly numerically challenging) Kendall distribution function $K$ is used (include.K=TRUE) or not (include.K=FALSE).

References

Hering, C. and Hofert, M. (2014). Goodness-of-fit tests for Archimedean copulas in high dimensions. Innovations in Quantitative Risk Management.

Examples

Run this code

## Sample and build pseudo-observations (what we normally have available)
## of a Clayton copula
tau <- 0.5
theta <- iTau(claytonCopula(), tau = tau)
d <- 5
cc <- claytonCopula(theta, dim = d)
set.seed(271)
n <- 1000
U <- rCopula(n, copula = cc)
X <- qnorm(U) # X now follows a meta-Gumbel model with N(0,1) marginals
U <- pobs(X) # build pseudo-observations

## Graphically check if the data comes from a meta-Clayton model
## with the transformation of Hering and Hofert (2014):
U.H <- htrafo(U, copula = cc) # transform the data
splom2(U.H, cex = 0.2) # looks good

## The same for an 'outer_nacopula' object
cc. <- onacopulaL("Clayton", list(theta, 1:d))
U.H. <- htrafo(U, copula = cc.)
splom2(U.H., cex = 0.2) # looks good

## What about a meta-Gumbel model?
## The parameter is chosen such that Kendall's tau equals (the same) tau
gc <- gumbelCopula(iTau(gumbelCopula(), tau = tau), dim = d)

## Plot of the transformed data (Hering and Hofert (2014)) to see the
## deviations from uniformity
U.H.. <- htrafo(U, copula = gc)
splom2(U.H.., cex = 0.2) # deviations visible

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