GOF Testing Transformations for Archimedean Copulas

Description

Compute the following goodness-of-fit (GOF) testing transformations,

Rosenblatt's transformation rtrafo()

is also of importance outside of GOF computations: The Rosenblatt transformation is used for computing conditional copulas and for sampling purposes. Currently, rtrafo() is applicable to elliptical and Archimedean copulas.

htrafo()

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

Usage

rtrafo(u, cop, j.ind = NULL, n.MC=0, inverse=FALSE, log=FALSE)
htrafo(u, cop, include.K=TRUE, n.MC=0, inverse=FALSE,
       method=eval(formals(qK)$method), u.grid, ...)

Arguments

Details

rtrafo

Given a d-dimensional random vector U following an Archimedean copula C with generator ψ, the conditional copula of U_j=u_j given U_1=u_1,..., U_{j-1}=u_{j-1} is given by

C(u_j | u_1,...,u_{j-1}) = (psi^{(j-1)}(sum(k=1.. j) psi^{(-1)}(u_k))) / (psi^{(j-1)}(sum(k=1..j-1) psi^{(-1)}(u_k))).

This formula is either evaluated with the exact derivatives or, if n.MC is positive, via Monte Carlo; see absdPsiMC.

Rosenblatt (1952) showed that U' ~ U[0,1]^m, where U'_1 = U_1, U'_2 = C(U_2 | U_1), ..., and U'_m = C(U_m | U_1,..., U_{m-1}).

rtrafo applies this transformation row-wise to u (with default m=d) and thus returns an n x m-matrix.

The inverse transformation (inverse=TRUE) applied to U[0,1]^d data is known as “conditional distribution method” for sampling.

Note that for the Clayton, the Gauss and the t copula, both the conditional copulas and their inverses are known explicitly and rtrafo() utilizes these explicit forms.

htrafo

Given a d-dimensional random vector U following an Archimedean copula C with generator ψ, Hering and Hofert (2014) showed that U'~U[0,1]^d, where

U'_j = ((psi^{-1}(U_1) + ... + psi^{-1}(U_j)) / (psi^{-1}(U_1) + ... + psi^{-1}(U_{j+1})))^j, j in {1,..., d-1}, U'_d = K(C(U)).

htrafo applies this transformation row-wise to u and thus returns either an n x d- or an n x (d-1)-matrix, depending on whether the last component U'_d which involves the (possibly numerically challenging) Kendall distribution function K is used (include.K=TRUE) or not (include.K=FALSE).

Value

htrafo() returns an n x d- or n x (d-1)-matrix (depending on whether include.K is TRUE or FALSE) containing the transformed input u.

rtrafo() returns an n x d-matrix containing the transformed input u.

Author(s)

Marius Hofert, Martin Maechler.

References

Genest, C., R<c3><83><c2><a9>millard, B., and Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics 44, 199–213.

Rosenblatt, M. (1952). Remarks on a Multivariate Transformation, The Annals of Mathematical Statistics 23, 3, 470–472.

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

Hofert, M., M<c3><83><c2><a4>chler, M., and McNeil, A. J. (2012). Likelihood inference for Archimedean copulas in high dimensions under known margins. Journal of Multivariate Analysis 110, 133–150.

See Also

gofCopula where both transformations are applied or emde where htrafo is applied.

Examples

tau <- 0.5
(theta <- copGumbel@iTau(tau)) # 2
(copG <- onacopulaL("Gumbel", list(theta, 1:5))) # d = 5

set.seed(1)
n <- 1000
x <- rnacopula(n, copG)
x <- qnorm(x) # 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-Gumbel model
## with the transformation of Hering and Hofert (2014):
u.h <- htrafo(u, cop=copG) # transform the data
pairs(u.h, gap=0, cex=0.2) # looks good

## with the transformation of Rosenblatt (1952):
u.r <- rtrafo(u, cop=copG) # transform the data
pairs(u.r, gap=0, cex=0.2) # looks good

## what about a meta-Clayton model?
## the parameter is chosen such that Kendall's tau equals (the same) tau
copC <- onacopulaL("Clayton", list(copClayton@iTau(tau), 1:5))

## plot of the transformed data (Hering and Hofert (2014)) to see the
## deviations from uniformity
u.H <- htrafo(u, cop=copC) # transform the data
pairs(u.H, gap=0, cex=0.2) # clearly visible

## plot of the transformed data (Rosenblatt (1952)) to see the
## deviations from uniformity
u.R <- rtrafo(u, cop=copC) # transform the data
pairs(u.R, gap=0, cex=0.2) # clearly visible

## rtrafo() for elliptical:
fN <- fitCopula(normalCopula(dim=ncol(u)), u) # fit a Gauss copula
pairs(rtrafo(u, cop=fN@copula), gap=0, cex=0.2) # visible but not so clearly
if(copula:::doExtras()) {
  f.t <- fitCopula(tCopula(dim=ncol(u)), u)
  tCop <- f.t@copula
} else {
  tCop <- tCopula(param = 0.685, df = 7, dim=ncol(u))
}
u.Rt <- rtrafo(u, cop=tCop) # transform with a fitted t copula
pairs(u.Rt, gap=0, cex=0.2) # *not* clearly visible

Results