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**

emle.Rd

```
\name{emle}
\title{Maximum Likelihood Estimators for (Nested) Archimedean Copulas}
\alias{emle}
\alias{.emle}
\description{
Compute (simulated) maximum likelihood estimators for (nested)
Archimedean copulas.
}
\usage{
emle(u, cop, n.MC=0, optimizer="optimize", method,
interval=initOpt(cop@copula@name),
start=list(theta=initOpt(cop@copula@name, interval=FALSE, u=u)),
\dots)
.emle(u, cop, n.MC=0,
interval=initOpt(cop@copula@name), \dots)
}
\arguments{
\item{u}{\eqn{n\times d}{n x d}-matrix of (pseudo-)observations (each
value in \eqn{[0,1]}) from the copula, with \eqn{n} the sample size
and \eqn{d} the dimension.}
\item{cop}{\code{\linkS4class{outer_nacopula}} to be estimated
(currently only non-nested, that is, \ifelse{latex}{Archi-medean}{Archimedean} copulas are admitted).}
\item{n.MC}{\code{\link{integer}}, if positive, \emph{simulated} maximum
likelihood estimation (SMLE) is used with sample size equal to
\code{n.MC}; otherwise (\code{n.MC=0}), MLE. In SMLE, the \eqn{d}th
generator derivative and thus the copula density is evaluated via
(Monte Carlo) simulation, whereas MLE uses the explicit formulas for
the generator derivatives; see the details below.
}
\item{optimizer}{a string or \code{NULL}, indicating the optimizer to
be used, where \code{NULL} means to use \code{\link{optim}} via the
standard \R function \code{\link[stats4]{mle}()} from package \pkg{stats4},
whereas the default, \code{"optimize"} uses \code{\link{optimize}} via
the \R function \code{\link[bbmle]{mle2}()} from package \pkg{bbmle}.}
\item{method}{only when \code{optimizer} is \code{NULL} or
\code{"optim"}, the method to be used for \code{\link{optim}}.}
\item{interval}{bivariate vector denoting the interval where
optimization takes place. The default is computed as described in
Hofert et al. (2011a).}
\item{start}{\code{\link{list}} of initial values, passed through.}
\item{\dots}{additional parameters passed to \code{\link{optimize}}.}
}
\details{
Exact formulas for the generator derivatives were derived in Hofert
et al. (2011b). Based on these formulas one can compute the
(log-)densities of the Archimedean copulas. Note that for some
densities, the formulas are numerically highly non-trivial to compute
and considerable efforts were put in to make the computations
numerically feasible even in large dimensions (see the source code of
the Gumbel copula, for example). Both MLE and SMLE showed good
performance in the simulation study conducted by Hofert et
al. (2011a) including the challenging 100-dimensional case.
Alternative estimators (see also \code{\link{enacopula}}) often used
because of their numerical feasibility, might break down in much
smaller dimensions.
Note: SMLE for Clayton currently faces serious numerical issues and is
due to further research. This is only interesting from a theoretical point
of view, since the exact derivatives are known and numerically non-critical
to evaluate.
}
\value{
\describe{
\item{emle}{
an \R object of class \code{"\link[bbmle:mle2-class]{mle2}"} (and
thus useful for obtaining confidence intervals) with the
(simulated) maximum likelihood estimator.}
\item{.emle}{\code{\link{list}} as returned by
\code{\link{optimize}()} including the maximum likelihood
estimator (does not confidence intervals but is typically faster).}
}
}
\author{Martin Maechler, Marius Hofert.}
\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[bbmle]{mle2}} from package \pkg{bbmle} and
\code{\link[stats4]{mle}} from \pkg{stats4} on which \code{mle2} is
modeled. \code{\link{enacopula}} (wrapper for different estimators).
\code{\link{demo}(opC-demo)} and \code{\link{demo}(GIG-demo)} for
examples of two-parameter families.
}
\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)
## Estimation
system.time(efm <- emle(U, cop))
summary(efm)
## Profile likelihood plot
pfm <- profile(efm)
(ci <- confint(pfm, level=0.95))
stopifnot(ci[1] <= theta, theta <= ci[2])
plot(pfm) # |z| against theta, |z| = sqrt(deviance)
plot(pfm, absVal=FALSE, # z against theta
show.points=TRUE) # showing how it's interpolated
## and show the true theta:
abline(v=theta, col="lightgray", lwd=2, lty=2)
axis(1, pos = 0, at=theta, label=expression(theta[0]))
## Plot of the log-likelihood, MLE and conf.int.:
logL <- function(x) -efm@minuslogl(x)
# == -sum(copGumbel@dacopula(U, theta=x, log=TRUE))
logL. <- Vectorize(logL)
I <- c(cop@copula@tauInv(0.1), cop@copula@tauInv(0.4))
curve(logL., from=I[1], to=I[2], xlab=quote(theta),
ylab="log-likelihood",
main="log-likelihood for Gumbel")
abline(v = c(theta, efm@coef), col="magenta", lwd=2, lty=2)
axis(1, at=c(theta, efm@coef), padj = c(-0.5, -0.8), hadj = -0.2,
col.axis="magenta", label= expression(theta[0], hat(theta)[n]))
abline(v=ci, col="gray30", lwd=2, lty=3)
text(ci[2], extendrange(par("usr")[3:4], f= -.04)[1],
"95\% conf. int.", col="gray30", adj = -0.1)
}
\keyword{models}
```

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