copula (version 0.999-1)

gtrafo: Goodness-of-fit testing transformations for (nested) Archimedean copulas

Description

htrafo() the transformation described in Hering and Hofert (2011).

rtrafo() the transformation of Rosenblatt (1952).

Usage

htrafo(u, cop, include.K=TRUE, n.MC=0, inverse=FALSE,
       method=formals(qK)$method, u.grid, ...)
rtrafo(u, cop, m=d, n.MC=0)

Arguments

u

$n\times d$-matrix of (pseudo-/copula-)observations (each value in $[0,1]$) from the copula cop based on which the transformation is carried out. Consider applying the function pobs fi

cop

the "outer_nacopula" with specified parameters based on which the transformation is computed (currently only Archimedean copulas are provided).

include.K

logical indicating whether the last component, involving the Kendall distribution function K, is used in htrafo.

m

integer between 2 and $d$ indicating the order up to which the conditional distributions are computed (the largest being the $m$th given all with index smaller than $m$).

n.MC

parameter n.MC for K (for htrafo) or for approximating the derivatives involved (for rtrafo).

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.

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.

  rtrafo() returns an $n\times d$-matrix containing the transformed input u.

Details

[object Object],[object Object]

References

Genest, C., Ré{e}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. (2011), Goodness-of-fit tests for Archimedean copulas in large dimensions, submitted.

Hofert, M., Mächler{Maechler}, M., and McNeil, A. J. (2011b), Likelihood inference for Archimedean copulas, submitted.

See Also

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

Examples

tau <- 0.5
(theta <- copGumbel@tauInv(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 (2011):
u.h <- htrafo(u, cop=copG) # transform the data
pairs(u.h, cex=0.2) # looks good

## with the transformation of Rosenblatt (1952):
u.r <- rtrafo(u, cop=copG) # transform the data
pairs(u.r, 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@tauInv(tau), 1:5))

## plot of the transformed data (Hering and Hofert (2011)) to see the
## deviations from uniformity
u.prime <- htrafo(u, cop=copC) # transform the data
pairs(u.prime, cex=0.2) # clearly visible

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