copula (version 1.1-7)

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.

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

## 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