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\name{EllipticalDependency}
\alias{EllipticalDependency}
\alias{ellipticalTau}
\alias{ellipticalRho}
\alias{ellipticalTailCoeff}
\alias{ellipticalTailPlot}
\title{Bivariate Elliptical Copulae}
\description{
A collection and description of functions to investigate
bivariate elliptical copulae.
\cr
Elliptical Copulae Functions:
\tabular{ll}{
\code{ellipticalTau} \tab Computes Kendall's tau for elliptical copulae, \cr
\code{ellipticalRho} \tab computes Spearman's rho for elliptical copulae, \cr
\code{ellipticalTailCoeff} \tab computes tail dependence for elliptical copulae, \cr
\code{ellipticalTailPlot} \tab plots tail dependence for elliptical copulae. }
}
\usage{
ellipticalTau(rho)
ellipticalRho(rho, param = NULL, type = ellipticalList(), subdivisions = 500)
ellipticalTailCoeff(rho, param = NULL, type = c("norm", "cauchy", "t"))
ellipticalTailPlot(param = NULL, type = c("norm", "cauchy", "t"),
tail = c("Lower", "Upper"))
}
\arguments{
\item{rho}{
[*ellipticalCopula] - \cr
is the numeric value setting the correlation strength, ranging
between minus one and one.
}
\item{param}{
[*ellipticalCopula][gfunc] - \cr
additional distributional parameters: for the Sudent-t distribution
this is "nu", for the Kotz distribution this is "r", and for the
Exponential Power distribution these are "r" and "s". If the
argument \code{param=NULL} then default values are taken. These are
for the Student-t \code{param=c(nu=4))}, for the Kotz distribution
\code{param=c(r=1))}, and for the exponential power distribution
\code{param=c(r=1,s=1)}. Note, that the Kotz and exponential power
copulae are independent of \code{r}, and that \code{r} only enters
the generator, the density, the probability and the quantile
functions.
}
\item{subdivisions}{
[ellipticalRho] - \cr
an integer value with the number of subdivisons in each direction
on the two dimensional unit square to compute the mean value of
Spearman's Rho. By default 500 subdivisions are used.
}
\item{tail}{
[ellipticalTailPlot] - \cr
a character string, either \code{"Upper"} or \code{"Lower"} denoting
which of the two tails should be displayed. By default the upper
tail dependence will be considered.
}
\item{type}{
[*ellipticalCopula][gfunc] - \cr
the type of the elliptical copula. A character string selected
from: "norm", "cauchy", "t", "logistic", "laplace", "kotz",
or "epower".
[*ellipticalSlider] - \cr
a character string which indicates what kind of plot should be
displayed, either a perspective plot if \code{type="persp"}, the
default value, or a contour plot if \code{type="contour"}.
}
}
\value{
\bold{Copula Functions:}
\cr\cr
The functions \code{[rpd]ellipticalCopula} return a numeric vector
of random variates, probabilities, or densities for the specified
copula computed at grid coordinates \code{u}|\code{v}.
\cr
The functions \code{[rpd]ellipticalSlider} display an interactive
graph of an perspective copula plot either for random variates,
probabilities or densities. Alternatively, an image underlayed
contour plot can be shown.
\cr
\bold{Copula Dependence Measures:}
\cr\cr
The functions \code{ellipticalTau} and \code{ellipticalRho} return
a numericc value for Kendall's Tau and Spearman's Rho.
\cr
\bold{Copula Tail Coefficient:}
\cr\cr
The function \code{ellipticalTailCoeff} returns the coefficient of
tail dependence for a specified copula. The function
\code{ellipticalTailPlot} displays a whole plot for the upper or
alternatively for the lower tail dependence as a function of
\code{u} for a set of nine \code{rho} values.
\cr
\bold{Copula Generator Function:}
\cr\cr
The function \code{gfunc} computes the generator function for the
specified copula, by default the normal copula. If the argument
\code{x} is missing, then the normalization constand lambda will
be returned, otherwise if \code{x} is specified the values for the
function \emph{g(x)} will be freturned. The selected type of copula
is added to the output as an attribute named \code{"control"}.
The function \code{gfuncSlider} allows to display interactively
the generator function, the marginal density, the marginal
probability, and the contours of the the bivariate density.
\cr
\bold{Copula Simulation and Parameter Fitting:}
\cr\cr
The function \code{ellipticalCopulaSim} returns a numeric two-column
matrix with randomly generated variates for the specified copula.
\cr
The function \code{ellipticalCopulaFit} returns a fit to empirical
data for the specified copula. The returned object is a list with
elements from the function \code{nlminb}.
}
\author{
Diethelm Wuertz for the Rmetrics \R-port.
}
\examples{
## [rp]ellipticalCopula -
# Default Normal Copula:
rellipticalCopula(10)
pellipticalCopula(10)
## [rp]ellipticalCopula -
# Student-t Copula Probability and Density:
u = grid2d(x = (0:25)/25)
pellipticalCopula(u, rho = 0.75, param = 4,
type = "t", output = "list")
d <- dellipticalCopula(u, rho = 0.75, param = 4,
type = "t", output = "list")
persp(d, theta = -40, phi = 30, col = "steelblue")
## ellipticalTau -
## ellipticalRho -
# Dependence Measures:
ellipticalTau(rho = -0.5)
ellipticalRho(rho = 0.75, type = "logistic", subdivisions = 100)
## ellipticalTailCoeff -
# Student-t Tail Coefficient:
ellipticalTailCoeff(rho = 0.25, param = 3, type = "t")
## gfunc -
# Generator Function:
plot(gfunc(x = 0:10), main = "Generator Function")
## ellipticalCopulaSim -
## ellipticalCopulaSim -
# Simualtion and Parameter Fitting:
rv <- ellipticalCopulaSim(n = 100, rho = 0.75)
ellipticalCopulaFit(rv)
}
\keyword{models}
|
