https://github.com/cran/nacopula
Tip revision: 161411bb86f97e5a8bd89091cd61d03a33c2761a authored by Martin Maechler on 06 February 2012, 00:00:00 UTC
version 0.8-0
version 0.8-0
Tip revision: 161411b
beta.Blomqvist.Rd
\name{beta.Blomqvist}
\title{Blomqvist's Beta for Archimedean Copula, Sample and Population}
\alias{beta.}
\alias{beta.hat}
\description{
Compute the population (\code{beta.()}) and sample
(\code{beta.hat()}) version of Blomqvist's beta for an Archimedean
copula.
See the reference below for definitions and formulas.
}
\usage{
beta.(cop, theta, d, scaling=FALSE)
beta.hat(u, scaling=FALSE)
}
\arguments{
\item{cop}{an Archimedean copula (of dimension \eqn{d}) to be
estimated.}
\item{theta}{copula parameter.}
\item{d}{dimension.}
\item{scaling}{logical, if true, the factors 2^(d-1)/(2^(d-1)-1) and
2^(1-d) in Blomqvist's beta are omitted.}
\item{u}{For \code{beta.hat}: (\eqn{n\times d}{n x d})-matrix of
d-dimensional observations distributed according to the copula.}
}
\value{
\describe{
\item{\code{beta.}:}{a number, being the population version of
Blomqvist's beta for the corresponding
\ifelse{latex}{Archi-medean}{Archimedean} copula;}
\item{\code{beta.hat}:}{a number, being the sample version of Blomqvist's beta for the given data.}
}
}
\references{
Schmid and Schmidt (2007),
Nonparametric inference on multivariate versions of Blomqvist's beta
and related measures of tail dependence,
\emph{Metrika} \bold{66}, 323--354.
}
\author{Marius Hofert}
\seealso{
\code{\linkS4class{acopula}}
}
\examples{
beta.(copGumbel, 2.5, d = 5)
d.set <- c(2:6, 8, 10, 15, 20, 30)
cols <- adjustcolor(colorRampPalette(c("red", "orange", "blue"),
space = "Lab")(length(d.set)), 0.8)
## AMH:
for(i in seq_along(d.set))
curve(Vectorize(beta.,"theta")(copAMH, x, d = d.set[i]), 0, .999999,
main = "Blomqvist's beta(.) for AMH",
xlab = expression(theta), ylab = expression(beta(theta, AMH)),
add=(i > 1), lwd=2, col=cols[i])
mtext("NB: d=2 and d=3 are the same")
legend("topleft", paste("d =",d.set), bty="n", lwd=2, col=cols)
## Gumbel:
for(i in seq_along(d.set))
curve(Vectorize(beta.,"theta")(copGumbel, x, d = d.set[i]), 1, 10,
main = "Blomqvist's beta(.) for Gumbel",
xlab = expression(theta), ylab = expression(beta(theta, Gumbel)),
add=(i > 1), lwd=2, col=cols[i])
legend("bottomright", paste("d =",d.set), bty="n", lwd=2, col=cols)
## Clayton:
for(i in seq_along(d.set))
curve(Vectorize(beta.,"theta")(copClayton, x, d = d.set[i]), 1e-5, 10,
main = "Blomqvist's beta(.) for Clayton",
xlab = expression(theta), ylab = expression(beta(theta, Gumbel)),
add=(i > 1), lwd=2, col=cols[i])
legend("bottomright", paste("d =",d.set), bty="n", lwd=2, col=cols)
## Joe:
for(i in seq_along(d.set))
curve(Vectorize(beta.,"theta")(copJoe, x, d = d.set[i]), 1, 10,
main = "Blomqvist's beta(.) for Joe",
xlab = expression(theta), ylab = expression(beta(theta, Gumbel)),
add=(i > 1), lwd=2, col=cols[i])
legend("bottomright", paste("d =",d.set), bty="n", lwd=2, col=cols)
## Frank:
for(i in seq_along(d.set))
curve(Vectorize(beta.,"theta")(copFrank, x, d = d.set[i]), 1e-5, 50,
main = "Blomqvist's beta(.) for Frank",
xlab = expression(theta), ylab = expression(beta(theta, Gumbel)),
add=(i > 1), lwd=2, col=cols[i])
legend("bottomright", paste("d =",d.set), bty="n", lwd=2, col=cols)
## Shows the numeric problems:
curve(Vectorize(beta.,"theta")(copFrank, x, d = 29), 35, 42, col="violet")
}
\keyword{multivariate}
\keyword{distribution}