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##### https://github.com/cran/nacopula
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
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Tip revision: 161411b
acopula-class.Rd
\name{acopula-class}
\Rdversion{1.1}
\title{Class "acopula" of Archimedean Copula Families}
\docType{class}
\alias{acopula-class}
\alias{acopula}
%
\alias{initialize,acopula-method}
\alias{show,acopula-method}
%
\description{
This class \code{"acopula"} of Archimedean Copula Families
is mainly used for providing objects of known Archimedean families
with all related functions.
}
\section{Objects from the Class}{
Objects can be created by calls of the form \code{new("acopula", ...)}.
For several well-known \ifelse{latex}{Archi-medean}{Archimedean}
copula families, the package \pkg{nacopula} already provides such
family objects.
}
\section{Slots}{
\describe{
\item{\code{name}:}{A string (class \code{"character"}) describing
the copula family, for example, "AMH" (or simply "A"), "Clayton" ("C"),
"Frank" ("F"), "Gumbel" ("G"), or "Joe" ("J").}
\code{NA} means \dQuote{unspecified}.}
\item{\code{psi}, \code{psiInv}:}{The (Archimedean) generator
\eqn{\psi}{psi} (with \eqn{\psi}{psi}(t)=\eqn{\exp}{exp}(-t) being the
generator of the independence copula) and its inverse
(\code{\link{function}}). \code{psiInv} has an optional argument
\code{log} which, if \code{TRUE} returns the logarithm of inverse of the
generator.}
\item{\code{psiDabs}:}{A \code{\link{function}} which computes the absolute
value of the derivative of the generator \eqn{\psi}{psi} for the given
parameter \code{theta} and of the given \code{degree} (defaults to 1).
Note that there is no informational loss by computing the absolute value
since the derivatives alternate in sign (the generator derivative
is simmply (-1)^\code{degree} times \code{psiDabs}).  The
number \code{n.MC} denotes the sample size for a Monte Carlo evaluation
approach.  If \code{n.MC} is zero (the default), the
generator derivatives are evaluated with their exact formulas.  The
optional parameter \code{log} (defaults to \code{FALSE}) indicates whether
or not the logarithmic value is returned.}
absolute value of the derivative of the generator inverse
(\code{psiInv()}) for the given parameter \code{theta}.  The optional
parameter \code{log} (defaults to \code{FALSE}) indicates whether
the logarithm of the absolute value of the first derivative of
\code{psiInv()} is returned.}
\item{\code{dDiag}:}{a \code{\link{function}} computing the density of the
diagonal of the Archimedean copula at \code{u} with parameter
\code{theta}.  The parameter \code{log} is as described before.}
of the Archimedean copula at \code{u} with parameter \code{theta}.  The
meanings of the parameters \code{n.MC} and \code{log} are as
described before.}
of the density with respect to the parameter \eqn{\theta}{theta}.}
\item{\code{paraInterval}:}{Either \code{\link{NULL}} or an object of class
\code{"\linkS4class{interval}"}, which is typically obtained from a call
\item{\code{paraConstr}:}{A function of \code{theta} returning
\code{TRUE} if and only if \code{theta} is a valid parameter value.  Note
that \code{paraConstr} is built automatically from the interval,
whenever the \code{paraInterval} slot is valid.
\code{TRUE} if and only if the two provided parameters \code{theta0} and
\code{theta1} satisfy the sufficient nesting condition for this family.}
\item{\code{V0}:}{A \code{\link{function}} which samples \code{n} random
variates from the distribution \eqn{F} with Laplace-Stieltjes transform
\eqn{\psi}{psi} and parameter \code{theta}.}
\item{\code{dV0}:}{A \code{\link{function}} which computes either the
probability mass function or the probability density function (depending
on the Archimedean family) of the distribution function whose
Laplace-Stieltjes transform equals the generator \eqn{\psi}{psi} at the
argument \code{x} (possibly a vector) for the given parameter
\code{theta}.  An optional argument \code{log} indicates whether
the logarithm of the mass or density is computed (defaults to
\code{FALSE}).}
\item{\code{V01}:}{A \code{\link{function}} which gets a vector of
realizations of \code{V0}, two parameters \code{theta0} and \code{theta1}
which satisfy the sufficient nesting condition, and which
returns a vector of the same length as \code{V0} with random variates from
the distribution function \eqn{F_{01}}{F01} with Laplace-Stieltjes
transform \eqn{\psi_{01}}{psi01} (see \code{dV01}) and parameters
\eqn{\theta_0=\,}{theta0=}\code{theta0},
\eqn{\theta_1=\,}{theta1=}\code{theta1}.}
\item{\code{dV01}:}{Similar to \code{dV0} with the difference being that
this \code{\link{function}} computes the probability mass or density
function for the Laplace-Stieltjes transform
\deqn{\psi_{01}(t;V_0) = \exp(-V_0\psi_0^{-1}(\psi_1(t))),
}{psi01(t;V0) = exp(-V0*psi0^{-1}(psi1(t))),}
corresponding to the distribution function \eqn{F_{01}}{F01}.

Arguments are the evaluation point(s) \code{x}, the value(s) \code{V0},
and the parameters \code{theta0} and \code{theta1}.  As for \code{dV0}, the
optional argument \code{log} can be specified (defaults to \code{FALSE}).
Note that if \code{x} is a vector, \code{V0} must either have
length one (in which case \code{V0} is the same for every component of
\code{x}) or \code{V0} must be of the same length as \code{x} (in which
case the components of \code{V0} correspond to the ones of \code{x}).}
\item{\code{tau}, \code{tauInv}:}{Compute Kendall's tau of the bivariate
Archimedean copula with generator \eqn{\psi}{psi} as a
\code{\link{function}} of \code{theta}, respectively, \code{theta} as a
function of Kendall's tau.}
\item{\code{lambdaL}, \code{lambdaU}, \code{lambdaLInv},
\code{lambdaUInv}:}{Compute the lower (upper) tail-dependence coefficient
of the bivariate Archimedean copula with generator \eqn{\psi}{psi} as a
\code{\link{function}} of \code{theta}, respectively, \code{theta} as a
function of the lower (upper) tail-dependence coefficient.}
}
For more details about Archimedean families, corresponding distributions and
properties, see the references.
}
\section{Methods}{
\describe{
\item{initialize}{\code{signature(.Object = "acopula")}: is used to
automatically construct the function slot \code{paraConstr}, when
the \code{paraInterval} is provided (typically via

\item{show}{\code{signature("acopula")}: compact overview of the copula.}
}
}
\author{Martin Maechler, Marius Hofert}
\seealso{
Specific provided copula family objects, for example,
\cr To access these, you may also use \code{\link{getAcop}}.

A \emph{nested} Archimedean copula \emph{without}
is a proper \ifelse{latex}{Archime-dean}{Archimedean} copula, and hence,
\code{\link{onacopula}()} can be used to construct a specific
parametrized Archimedean copula; see the example below.
}
\references{% -> ./copFamilies.Rd
See those of the families, for example, \code{\link{copGumbel}}.
}
\examples{
## acopula class information
showClass("acopula")

## Information and structure of Clayton copulas
copClayton
str(copClayton)

## What are admissible parameters for Clayton copulas?
copClayton@paraInterval

## Can two Clayton copulas with parameters theta0 and theta1 be nested?
## Case 1: theta0 = 3, theta1 = 2
copClayton@nestConstr(theta0 = 3, theta1 = 2)
## -> FALSE as the sufficient nesting criterion is not fulfilled
## Case 2: theta0 = 2, theta1 = 3
copClayton@nestConstr(theta0 = 2, theta1 = 3) # TRUE

## For more examples, see  help("acopula-families")
}
\keyword{multivariate}
\keyword{classes}


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