gnacopula: Goodness-of-fit Testing for (Nested) Archimedean Copulas

Description

gnacopula() conducts a goodness-of-fit test for the given (\(H_0\)-)copula cop based on the (copula-)data u.

NOTE: gnacopula() is deprecated, call gofCopula() instead.

Usage

gnacopula(u, cop, n.bootstrap,
          estim.method = eval(formals(enacopula)$method),
          include.K=TRUE, n.MC=0, trafo=c("Hering.Hofert", "Rosenblatt"),
          method=eval(formals(gofTstat)$method), verbose=TRUE, ...)

Value

gnacopula returns an R object of class "htest". This object contains a list with the bootstrap results including the components

p.value:: the bootstrapped p-value;
statistic:: the value of the test statistic computed for the data u;
data.name:: the name of u;
method:: a character describing the goodness-of-fit test applied;
estimator:: the estimator computed for the data u;
bootStats:: a list with component estimator containing the estimators for all bootstrap replications and component statistic containing the values of the test statistic for each bootstrap replication.

Arguments

u

\(n\times d\)-matrix of values in \([0,1]\); should be (pseudo-/copula-)observations from the copula to be tested. Consider applying the function pobs() first in order to obtain u.

cop

\(H_0\)-"outer_nacopula" with specified parameters to be tested for (currently only Archimedean copulas are provided).

n.bootstrap

positive integer specifying the number of bootstrap replicates.

estim.method

character string determining the estimation method; see enacopula(). We currently only recommend the default "mle" (or maybe "smle").

include.K

logical indicating whether the last component, involving the Kendall distribution function K(), is used in the transformation htrafo() of Hering and Hofert (2011). Note that this only applies to trafo="Hering.Hofert".

n.MC

parameter n.MC for htrafo() (and thus for K()) if trafo="Hering.Hofert" and for cCopula() if trafo="Rosenblatt".

trafo

a character string specifying the multivariate transformation performed for goodness-of-fit testing, which has to be one (or a unique abbreviation) of

"Hering.Hofert": for the multivariate transformation of Hering and Hofert (2011); see htrafo().

"Rosenblatt"

for the multivariate transformation of Rosenblatt (1952); see cCopula().

method

a character string specifying the goodness-of-fit test statistic to be used; see gofTstat().

verbose

if TRUE, the progress of the bootstrap is displayed via txtProgressBar.

...

additional arguments passed to enacopula().

Details

The function gnacopula() performs a parametric bootstrap for the goodness-of-fit test specified by trafo and method. The transformation given by trafo specifies the multivariate transformation which is first applied to the (copula-) data u (typically, the pseudo-observations are used); see htrafo() or cCopula() for more details. The argument method specifies the particular goodness-of-fit test carried out, which is either the Anderson-Darling test for the univariate standard uniform distribution (for method="AnChisq" or method="AnGamma") in a one-dimensional setup or the tests described in Genest et al. (2009) for the multivariate standard uniform distribution directly in a multivariate setup. As estimation method, the method provided by estim.method is used.

Note that a finite-sample correction is made when computing p-values; see gofCopula() for details.

A word of warning: Do work carefully with the variety of different goodness-of-fit tests that can be performed with gnacopula(). For example, among the possible estimation methods at hand, only MLE is known to be consistent (under conditions to be verified). Furthermore, for the tests based on the Anderson-Darling test statistic, it is theoretically not clear whether the parametric bootstrap converges. Consequently, the results obtained should be treated with care. Moreover, several estimation methods are known to be prone to numerical errors (see Hofert et al. (2013)) and are thus not recommended to be used in the parametric bootstrap. A warning is given if gnacopula() is called with a method not being MLE.

References

Genest, C., Rémillard, 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.