Revision **161411bb86f97e5a8bd89091cd61d03a33c2761a** authored by Martin Maechler on **06 February 2012, 00:00:00 UTC**, committed by Gabor Csardi on **06 February 2012, 00:00:00 UTC**

Tip revision: **161411bb86f97e5a8bd89091cd61d03a33c2761a** authored by ** Martin Maechler ** on **06 February 2012, 00:00:00 UTC**

**version 0.8-0**

Tip revision: **161411b**

enacopula.Rd

```
\name{enacopula}
\alias{enacopula}
\title{Estimation Procedures for (Nested) Archimedean Copulas}
\description{
A set of ten different estimators, currently for one-parameter
Archimedean copulas, of possibly quite high dimensions.
}
\usage{
enacopula(u, cop,
method = c("mle", "smle", "dmle",
"mde.chisq.CvM", "mde.chisq.KS",
"mde.gamma.CvM", "mde.gamma.KS",
"tau.tau.mean", "tau.theta.mean", "beta"),
n.MC = if (method == "smle") 10000 else 0,
interval = initOpt(cop@copula@name),
xargs = list(), \dots)
}
\arguments{
\item{u}{\eqn{n\times d}{n x d}-matrix of (pseudo-)observations (each
value in \eqn{[0,1]}) from the copula to be estimated, where \eqn{n} denotes
the sample size and \eqn{d} the dimension. Consider applying the
function \code{\link{pobs}} first in order to obtain \code{u}.}
\item{cop}{\code{\linkS4class{outer_nacopula}} to be estimated
(currently only Archimedean copulas are provided).}
\item{method}{a \code{\link{character}} string specifying the
estimation method to be used, which has to be one (or a unique
abbreviation) of
\describe{
\item{\code{"mle"}}{maximum likelihood estimator (MLE) computed via \code{\link{.emle}}.}
\item{\code{"smle"}}{simulated maximum likelihood estimator (SMLE)
computed with the function \code{\link{.emle}}, where
\code{n.MC} gives the Monte Carlo sample size.}
\item{\code{"dmle"}}{MLE based on the diagonal (DMLE); see \code{\link{edmle}}.}
\item{\code{"mde.chisq.CvM"}}{minimum distance estimator based
on the chisq distribution and Cram\enc{é}{e}r-von Mises distance; see \code{\link{emde}}.}
\item{\code{"mde.chisq.KS"}}{minimum distance estimation based on
the chisq distribution and Kolmogorov-Smirnov distance; see \code{\link{emde}}.}
\item{\code{"mde.gamma.CvM"}}{minimum distance estimation based on
the Erlang distribution and Cram\enc{é}{e}r-von Mises distance; see \code{\link{emde}}.}
\item{\code{"mde.gamma.KS"}}{minimum distance estimation based on
the Erlang distribution and Kolmogorov-Smirnov distance; see \code{\link{emde}}.}
\item{\code{"tau.tau.mean"}}{averaged pairwise Kendall's tau estimator}
\item{\code{"tau.theta.mean"}}{average of pairwise Kendall's tau estimators}
\item{\code{"beta"}}{multivariate Blomqvist's beta estimator}
}
}
\item{n.MC}{only for \code{method = "smle"}: \code{\link{integer}},
sample size for simulated maximum likelihood estimation.}
\item{interval}{bivariate vector denoting the interval where
optimization takes place. The default is computed as described in
Hofert et al. (2011a). Used for all methods except
\code{"tau.tau.mean"} and \code{"tau.theta.mean"}.}
\item{xargs}{list of additional arguments for the chosen estimation method.}
\item{\dots}{additional arguments passed to \code{\link{optimize}}.}
}
\details{
\code{\link{enacopula}} serves as a wrapper for the different
implemented estimators and provides a uniform framework to utilize
them. For more information, see the single estimators as given in the
section \sQuote{See Also}.
Note that Hofert, \enc{Mächler}{Maechler}, and McNeil (2011a) compared these
estimators. Their findings include a rather poor performance and numerically
challenging problems of some of these estimators. In particular, the
estimators obtained by \code{method="mde.gamma.CvM"},
\code{method="mde.gamma.KS"}, \code{method="tau.theta.mean"}, and
\code{method="beta"} should be used with care (or not at all). Overall, MLE
performed best (by far).
}
\value{
the estimated parameter, \eqn{\hat{\theta}}{hat(theta)}, that is, currently a
number as only one-parameter Archimedean copulas are considered.
}
\author{Marius Hofert, Martin Maechler.}
\references{
Hofert, M., \enc{Mächler}{Maechler}, M., and McNeil, A. J. (2011a).
Estimators for Archimedean copulas in high dimensions: A comparison,
to be submitted.
Hofert, M., \enc{Mächler}{Maechler}, M., and McNeil, A. J. (2011b).
Likelihood inference for Archimedean copulas,
submitted.
}
\seealso{
\code{\link{emle}} which returns an object of \code{"\linkS4class{mle}"}
providing useful methods not available for other estimators.
\code{\link{demo}(opC-demo)} and \code{\link{demo}(GIG-demo)} for
examples of two-parameter families.
\code{\link{edmle}} for the diagonal maximum likelihood estimator.
\code{\link{emde}} for the minimum distance estimators.
\code{\link{etau}} for the estimators based on Kendall's tau.
\code{\link{ebeta}} for the estimator based on Blomqvist's beta.
}
\examples{
tau <- 0.25
(theta <- copGumbel@tauInv(tau)) # 4/3
d <- 20
(cop <- onacopulaL("Gumbel", list(theta,1:d)))
set.seed(1)
n <- 200
U <- rnacopula(n, cop)
(meths <- eval(formals(enacopula)$method)) # "mle", "smle", ...
fun <- function(meth, u, cop, theta){
run.time <- system.time(val <- enacopula(u, cop=cop, method=meth))
list(value=val, error=val-theta, utime.ms=1000*run.time[[1]])
}
(res <- sapply(meths, fun, u=U, cop=cop, theta=theta))
}
\keyword{models}
```

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