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{$HeadURL: svn+ssh://ruckdeschel@svn.r-forge.r-project.org/svnroot/distr/pkg/distr/vignettes/newDistributions-knitr.Rnw $}
{$LastChangedDate: 2019-03-01 17:00:34 +0100 (Fr, 01 Mrz 2019) $}
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\svnRegisterAuthor{ruckdeschel}{Peter Ruckdeschel}
\svnRegisterAuthor{stamats}{Matthias Kohl}
\svnRegisterAuthor{florian}{Florian Camphausen}
\svnRegisterAuthor{stabla}{Thomas Stabla}
\svnRegisterAuthor{anhuel}{Anja H{\"u}ller}
\svnRegisterAuthor{ifrin}{Eleonara Feist}
\svnRegisterAuthor{jdospina}{Juan David Ospina}
\svnRegisterAuthor{kowzar}{Kouros Owzar}
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pdftitle={How to generate new distributions in packages distr, distrEx},%
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\markboth{\sl How to generate new distributions in packages ``{\tt distr}'', ``{\tt distrEx}''}%
{\sl How to generate new distributions in packages ``{\tt distr}'', ``{\tt distrEx}''}
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\title{How to generate new distributions in packages \pkg{distr}, \pkg{distrEx}}
%,version \pkgExversion}
\author{\small Peter Ruckdeschel\thanks{Universit\"at Oldenburg, Oldenburg}
\\[-.5ex]
\small Matthias Kohl\thanks{FH Furtwangen}
\smallskip\\
\small Institut f\"ur Mathematik\\[-0.5ex]
\small Fakult\"at V - Mathematik und Naturwissenschaften\\[-.5ex]
\small Carl von Ossietzky Universit\"at Oldenburg\\[-0.5ex]
\small PObox 2503\\[-.5ex]
\small 26111 Oldenburg (Oldb)\\[-.5ex]
\small Germany\\
\small e-Mail: {\small \tt peter.ruckdeschel@uni-oldenburg.de}\medskip\\
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\maketitle
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\begin{abstract}
% -------------------------------------------------------------------------------
In this vignette, we give short examples how to produce new
distributions in packages \pkg{distr} and \pkg{distrEx}.
This vignette refers to package version~\pkgversion.
% -------------------------------------------------------------------------------
\end{abstract}
% -------------------------------------------------------------------------------
Basically there are three ways to produce new
distributions in packages \pkg{distr} and \pkg{distrEx}:
\begin{enumerate}
\item automatic generation of single distribution objects by arithmetics and the like
\item using generating functions to produce single distribution objects
\item defining new distribution classes / doing it from scratch
\end{enumerate}
We will give short examples of all three of them.
% -------------------------------------------------------------------------------
\section{Automatic generation by arithmetics and the like}
% -------------------------------------------------------------------------------
We have made available quite general arithmetical operations to our distribution 
objects, generating new image distribution objects automatically. As an example, try
<<Prepa0, include=FALSE, results="hide">>=
require(distr)
@
<<Prepa, echo=FALSE, results="asis">>=
## preparation: set option withSweave to TRUE
require(distr)
distroptions(withSweave = TRUE)
options(width=70)
@
%
<<exam1, eval = TRUE, fig.width=8.0, fig.height=6.5>>=
require(distr)
N <- Norm(mean = 2, sd = 1.3)
P <- Pois(lambda = 1.2)
Z <- 2*N + 3 + P
Z
plot(Z, panel.first = grid(), lwd=3)
p(Z)(0.4)
q(Z)(0.3)
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
Zs <- r(Z)(50)
Zs
@
\par
\noindent{\bf Comment:}\\
Let \code{N} an object of class \code{"Norm"} with parameters  \code{mean=2},
\code{sd=1.3} and let \code{P}  an object of class \code{"Pois"} with parameter 
\code{lambda=1.2}. Assigning to \code{Z} the expression \code{2*N+3+P}, a 
new distribution object is generated ---of class \code{"AbscontDistribution"} in 
our case--- so that identifying \code{N}, \code{P}, \code{Z} with random 
variables distributed according to {\tt N}, {\tt P}, {\tt Z}, 
${\cal L}({\tt Z})={\cal L}(2*{\tt N}+3+{\tt P})$,  and writing \code{p(Z)(0.4)}  
we get $P(Z\leq 0.4)$, \code{ q(Z)(0.3)}  the $30\%$-quantile of {\tt Z},
and with \code{r(Z)(50)} we generate $50$ pseudo random numbers distributed 
according to {\tt Z}, while the \code{plot} command generates the above figure.\\

In the environments of \texttt{RStudio}, see \url{https://www.rstudio.com/} and
\texttt{Jupyter IRKernel}, see \url{https://github.com/IRkernel/IRkernel},
calls to \code{q} are caught away from standard {\sf R} evaluation and are treated
in a non-standard way. This non-standard evaluation in particular throws
errors at calls to our accessor methods \code{q} to slot \code{q} of the
respective distribution object. To amend this, from version 2.6 on,
we provide function \code{q.l} (for left-continuous quantile function)
as alias to our accessors \code{q}, so that all our package functionality also
becomes available in \texttt{RStudio} and \texttt{IRKernel}.\\

There are caveats to take care about; for details refer to the (larger) vignette
{\tt distr} in package \pkg{distrDoc}.
% -------------------------------------------------------------------------------
\section{Using generating functions}
% -------------------------------------------------------------------------------
If you want to generate a single distribution object (without any particular parameter)
generating functions are the method of choice:\\


Objects of classes \code{LatticeDistribution} resp.\ 
\code{DiscreteDistribution}, 
\code{AbscontDistribution},  may be generated using the generating functions
\code{LatticeDistribution()} resp.\ \code{DiscreteDistribution()}
resp.\ \code{AbscontDistribution()}; see also
the corresponding help.\\

E.g., to produce a discrete distribution with
support $(1,5,7,21)$ with corresponding probabilities $(0.1,0.1,0.6,0.2)$
we may write
<<DiscrDist, eval = TRUE>>=
D <- DiscreteDistribution(supp = c(1,5,7,21), prob = c(0.1,0.1,0.6,0.2))
D
plot(D, panel.first = grid(lwd=2), lwd = 3)
@
%
and to generate an absolutely continuous distribution with density proportional
to $e^{-|x|^3}$, we write
<<AbscDist, eval = TRUE>>=
AC <- AbscontDistribution(d = function(x) exp(-abs(x)^3), withStand = TRUE)
AC
plot(AC, panel.first = grid(lwd=2), lwd = 3)
@
%
% -------------------------------------------------------------------------------
\section{Doing it from scratch}
% -------------------------------------------------------------------------------
If you would like to create new parametric distributions, using already 
implemented {\tt r}, {\tt d}, {\tt p}, and {\tt q} functions
(e.g.\ implementing additional distributions realized in another 
\href{https://cran.r-project.org}{\tt CRAN} package),
you should probably envisage introducing new distribution {\tt S4} (sub-)classes
and hence better look at the implementation of some discrete and 
continuous parametric distribution classes in package \pkg{distr}.

\noindent{\small Hint: download the {\tt .tar.gz} file; extract it to some {\tt temp} 
folder; look at subdirectories {\tt R} and {\tt man}}\smallskip\\

The general procedure is as follows

\begin{enumerate}
\item introduce a  new subclass of  class \code{Parameter}
\item introduce a new subclass of  \code{LatticeDistribution}/%
\code{DiscreteDistribution} (if discrete)
 or of class \code{AbscontDistribution} (if continuous).
\item define accessor and replacement functions for the ``slots'' of the
parameter (e.g.\ \code{"size"} and \code{"prob"} in the binomial case), 
possibly with new generics
\item (possibly) define a validity function
\item define a generating function
\item if existing, define particular convolution methods or similar particular
methods for this new distribution class
\item create {\tt .Rd} files for the
\begin{itemize}
      \item parameter class
      \item distribution class
\end{itemize}
\item if analytic expressions are available, define particular \code{E}-, \code{var}-, 
\code{skewness}-, and \code{kurtosis}-methods
     and if so, also document\footnote{%
%
this is new, because so far, all \code{E}-, \code{var}-, 
\code{skewness}-, and \code{kurtosis}-methods for ``basic'' 
distributions are documented in the \pkg{distrEx} documentation to 
\code{E}, \code{var}, \ldots, but this would not be operational
any longer for new derived classes, possibly defined in other, new packages
%
     } the corresponding methods in
     the distribution class {\tt .Rd} file\\
\end{enumerate}
    
Let's go through the steps in the example case of the Binomial implementation
in packages \pkg{distr} and \pkg{distrEx}:

\begin{enumerate}
%
\item in \pkg{distr}, see source in \file{R/AllClasses.R},  
%
<<AllClass1, results="asis", echo=TRUE>>=
## Class: BinomParameter
setClass("BinomParameter",
          representation = representation(size = "numeric", prob = "numeric"),
          prototype = prototype(size = 1, prob = 0.5, name =
                      gettext("Parameter of a Binomial distribution")
                      ),
          contains = "Parameter"
          )
@
%
\item in \pkg{distr}, see source in \file{R/AllClasses.R},  
%
<<AllClass2, results="asis", echo=TRUE>>=
## Class: binomial distribution
setClass("Binom",
          prototype = prototype(
                      r = function(n){ rbinom(n, size = 1,prob = 0.5) },
                      d = function(x, log = FALSE){
                              dbinom(x, size = 1, prob = 0.5, log = log)
                                          },
                      p = function(q, lower.tail = TRUE, log.p = FALSE ){
                              pbinom(q, size = 1, prob = 0.5,
                                     lower.tail = lower.tail, log.p = log.p)
                                          },
                      q = function(p, lower.tail = TRUE, log.p = FALSE ){
                              qbinom(p, size = 1, prob = 0.5,
                                     lower.tail = lower.tail, log.p = log.p)
                                          },
                      img = new("Naturals"),
                      param = new("BinomParameter"),
                      support = 0:1,
                      lattice = new("Lattice",
                                pivot = 0, width = 1, Length = 2, name =
                                gettext(
                                  "lattice of a Binomial distribution"
                                       )
                                ),
                     .logExact = TRUE,
                     .lowerExact = TRUE
                      ),
          contains = "LatticeDistribution"
          )
@
%
\item in \pkg{distr}, see source in \file{R/BinomialDistribution.R},  
%
<<BinomDist1, results="asis", echo=TRUE>>=
## Access Methods
setMethod("size", "BinomParameter", function(object) object@size)
setMethod("prob", "BinomParameter", function(object) object@prob)
## Replace Methods
setReplaceMethod("size", "BinomParameter",
                  function(object, value){ object@size <- value; object})
setReplaceMethod("prob", "BinomParameter",
                  function(object, value){ object@prob <- value; object})
@
%
and \file{R/AllGenerics}, 
<<AllGenerics, results="asis", echo=TRUE>>=
if(!isGeneric("size"))
   setGeneric("size", function(object) standardGeneric("size"))
if(!isGeneric("prob"))
   setGeneric("prob", function(object) standardGeneric("prob"))
@
%
\item in \pkg{distr}, see source in \file{R/BinomialDistribution.R},  
%
<<BinomDist2, results="asis", echo=TRUE>>=
setValidity("BinomParameter", function(object){
  if(length(prob(object)) != 1)
    stop("prob has to be a numeric of length 1")
  if(prob(object) < 0)
    stop("prob has to be in [0,1]")
  if(prob(object) > 1)
    stop("prob has to be in [0,1]")
  if(length(size(object)) != 1)
    stop("size has to be a numeric of length 1")
  if(size(object) < 1)
    stop("size has to be a natural greater than 0")
  if(!identical(floor(size(object)), size(object)))
    stop("size has to be a natural greater than 0")
  else return(TRUE)
})
@
%
\item in \pkg{distr}, see source in \file{R/BinomialDistribution.R},  
%
<<BinomDist3, results="asis", echo=TRUE>>=
Binom <- function(size = 1,prob = 0.5) new("Binom", size = size, prob = prob)
@
%
\item in \pkg{distr}, see source in \file{R/BinomialDistribution.R},  
%
<<BinomDist4, results="asis", echo=TRUE>>=
## Convolution for two binomial distributions Bin(n1,p1) and Bin(n2,p2)
## Distinguish cases
## p1 == p2 und p1 != p2


setMethod("+", c("Binom","Binom"),
          function(e1,e2){
            newsize <- size(e1) + size(e2)

            if(isTRUE(all.equal(prob(e1),prob(e2))))
               return(new("Binom", prob = prob(e1), size = newsize,
                          .withArith = TRUE))

            return(as(e1, "LatticeDistribution") + e2)
          })
@
%
\item in \pkg{distr}, see sources in
%
\begin{itemize}
%
\item\file{man/BinomParameter-class.Rd}
%
\begin{lstlisting}[style = Rdstyle]
\name{BinomParameter-class}
\docType{class}
\alias{BinomParameter-class}
\alias{initialize,BinomParameter-method}

\title{Class "BinomParameter"}
\description{ The parameter of a binomial distribution, used by Binom-class}
\section{Objects from the Class}{
Objects can be created by calls of the form
      \code{new("BinomParameter", prob, size)}.
Usually an object of this class is not needed on its own, it is generated
automatically when an object of the class Binom
is instantiated.
}
\section{Slots}{
  \describe{
    \item{\code{prob}}{Object of class \code{"numeric"}:
          the probability of a binomial distribution }
    \item{\code{size}}{Object of class \code{"numeric"}:
          the size of a binomial distribution }
    \item{\code{name}}{Object of class \code{"character"}:
          a name / comment for the parameters }
  }
}
\section{Extends}{
Class \code{"Parameter"}, directly.
}
\section{Methods}{
  \describe{
    \item{initialize}{\code{signature(.Object = "BinomParameter")}:
          initialize method }
    \item{prob}{\code{signature(object = "BinomParameter")}: returns the slot
          \code{prob} of the parameter of the distribution }
    \item{prob<-}{\code{signature(object = "BinomParameter")}: modifies the slot
          \code{prob} of the parameter of the distribution }
    \item{size}{\code{signature(object = "BinomParameter")}: returns the slot
          \code{size} of the parameter of the distribution }
    \item{size<-}{\code{signature(object = "BinomParameter")}: modifies the slot
          \code{size} of the parameter of the distribution}
  }
}

\author{
  Thomas Stabla \email{statho3@web.de},\cr
  Florian Camphausen \email{fcampi@gmx.de},\cr
  Peter Ruckdeschel \email{peter.ruckdeschel@uni-oldenburg.de},\cr
  Matthias Kohl \email{Matthias.Kohl@stamats.de}
  }


\seealso{
\code{\link{Binom-class}}
\code{\link{Parameter-class}}
}

\examples{
W <- new("BinomParameter",prob=0.5,size=1)
size(W) # size of this distribution is 1.
size(W) <- 2 # size of this distribution is now 2.
}
\keyword{distribution}
\concept{parameter}
\concept{Binomial distribution}
\concept{S4 parameter class}
\end{lstlisting}

%
\item\file{man/Binom-class.Rd}
\begin{lstlisting}[style = Rdstyle]
\name{Binom-class}
\docType{class}
\alias{Binom-class}
\alias{Binom}
\alias{initialize,Binom-method}

\title{Class "Binom" }
\description{The binomial distribution with \code{size} \eqn{= n}, by default
  \eqn{=1}, and
  \code{prob} \eqn{= p}, by default \eqn{=0.5}, has density
  \deqn{p(x) = {n \choose x} {p}^{x} {(1-p)}^{n-x}}{
    p(x) = choose(n,x) p^x (1-p)^(n-x)}
  for \eqn{x = 0, \ldots, n}.

  C.f.\code{\link[stats:Binomial]{rbinom}}
}
\section{Objects from the Class}{
Objects can be created by calls of the form \code{Binom(prob, size)}.
This object is a binomial distribution.
}
\section{Slots}{
  \describe{
    \item{\code{img}}{Object of class \code{"Naturals"}: The space of the
     image of this distribution has got dimension 1 and the
     name "Natural Space". }
    \item{\code{param}}{Object of class \code{"BinomParameter"}: the parameter
          of this distribution (\code{prob}, \code{size}), declared at its
          instantiation }
    \item{\code{r}}{Object of class \code{"function"}: generates random
          numbers (calls function \code{rbinom}) }
    \item{\code{d}}{Object of class \code{"function"}: density function (calls
          function \code{dbinom}) }
    \item{\code{p}}{Object of class \code{"function"}: cumulative function
          (calls function \code{pbinom}) }
    \item{\code{q}}{Object of class \code{"function"}: inverse of the
           cumulative function (calls function \code{qbinom}).
    The quantile is defined as the smallest value x such that F(x) >= p, where
            F is the cumulative function. }
    \item{\code{support}}{Object of class \code{"numeric"}: a (sorted)
            vector containing the support of the discrete density function}
    \item{\code{.withArith}}{logical: used internally to issue warnings as to
            interpretation of arithmetics}
    \item{\code{.withSim}}{logical: used internally to issue warnings as to
          accuracy}
    \item{\code{.logExact}}{logical: used internally to flag the case where
    there are explicit formulae for the log version of density, cdf, and
    quantile function}
    \item{\code{.lowerExact}}{logical: used internally to flag the case where
    there are explicit formulae for the lower tail version of cdf and quantile
    function}
    \item{\code{Symmetry}}{object of class \code{"DistributionSymmetry"};
     used internally to avoid unnecessary calculations.}
  }
}
\section{Extends}{
Class \code{"DiscreteDistribution"}, directly.\cr
Class \code{"UnivariateDistribution"}, by class \code{"DiscreteDistribution"}.\cr
Class \code{"Distribution"}, by class \code{"DiscreteDistribution"}.
}
\section{Methods}{
  \describe{
    \item{+}{\code{signature(e1 = "Binom", e2 = "Binom")}: For two binomial
             distributions with equal probabilities the exact convolution
             formula is implemented thereby improving the general numerical
             accuracy.}
    \item{initialize}{\code{signature(.Object = "Binom")}: initialize method }
    \item{prob}{\code{signature(object = "Binom")}: returns the slot \code{prob}
             of the parameter of the distribution }
    \item{prob<-}{\code{signature(object = "Binom")}: modifies the slot
             \code{prob} of the parameter of the distribution }
    \item{size}{\code{signature(object = "Binom")}: returns the slot \code{size}
             of the parameter of the distribution }
    \item{size<-}{\code{signature(object = "Binom")}: modifies the slot
             \code{size} of the parameter of the distribution }
  }
}


\author{
  Thomas Stabla \email{statho3@web.de},\cr
  Florian Camphausen \email{fcampi@gmx.de},\cr
  Peter Ruckdeschel \email{peter.ruckdeschel@uni-oldenburg.de},\cr
  Matthias Kohl \email{Matthias.Kohl@stamats.de}
  }


\seealso{
\code{\link{BinomParameter-class}}
\code{\link{DiscreteDistribution-class}}
\code{\link{Naturals-class}}
\code{\link[stats:Binomial]{rbinom}}
}
\examples{
B <- Binom(prob=0.5,size=1) # B is a binomial distribution with prob=0.5 and size=1.
r(B)(1) # # one random number generated from this distribution, e.g. 1
d(B)(1) # Density of this distribution is  0.5 for x=1.
p(B)(0.4) # Probability that x<0.4 is 0.5.
q(B)(.1) # x=0 is the smallest value x such that p(B)(x)>=0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
size(B) # size of this distribution is 1.
size(B) <- 2 # size of this distribution is now 2.
C <- Binom(prob = 0.5, size = 1) # C is a binomial distribution with prob=0.5 and size=1.
D <- Binom(prob = 0.6, size = 1) # D is a binomial distribution with prob=0.6 and size=1.
E <- B + C # E is a binomial distribution with prob=0.5 and size=3.
F <- B + D # F is an object of class LatticeDistribution.
G <- B + as(D,"DiscreteDistribution") ## DiscreteDistribution
}
\keyword{distribution}
\concept{discrete distribution}
\concept{lattice distribution}
\concept{Binomial family}
\concept{Binomial distribution}
\concept{S4 distribution class}
\concept{generating function}
\end{lstlisting}

%
\item {\footnotesize you could have: \file{man/Binom.Rd}
      for the generating function; in the Binomial case, documentation is in
      \file{Binom-class.Rd}; but in case of the Gumbel distribution,
      in package \pkg{RobExtremes}, there is such an extra {\tt .Rd} file}
%
\end{itemize}
%
\item in \pkg{distrEx}, see sources in
%
<<Prepa2, echo=FALSE, results="asis">>=
## preparation: set option withSweave to TRUE
require(distrEx)
@
%
\begin{itemize}
%
\item\file{Expectation.R}, 
<<Expect, results="asis", echo=TRUE>>=
setMethod("E", signature(object = "Binom",
                         fun = "missing",
                         cond = "missing"),
    function(object, low = NULL, upp = NULL, ...){
    if(!is.null(low)) if(low <= min(support(object))) low <- NULL
    if(!is.null(upp)) if(upp >= max(support(object))) upp <- NULL
    if(is.null(low) && is.null(upp))
        return(size(object)*prob(object))
    else{
        if(is.null(low)) low <- -Inf
        if(is.null(upp)) upp <- Inf
        if(low == -Inf){
           if(upp == Inf) return(size(object)*prob(object))
           else return(m1df(object, upper = upp, ...))
        }else{
           E1 <- m1df(object, upper = low, ...)
           E2 <- if(upp == Inf)
                    size(object)*prob(object) else m1df(object, upper = upp, ...)
           return(E2-E1)
        }
    }
   })
@
%
\item\file{Functionals.R}, 
<<var, results="asis", echo=TRUE>>=
setMethod("var", signature(x = "Binom"),
    function(x,...){
    dots <- match.call(call = sys.call(sys.parent(1)),
                       expand.dots = FALSE)$"..."
    fun <- NULL; cond <- NULL; low <- NULL; upp <- NULL
    if(hasArg(low)) low <- dots$low
    if(hasArg(upp)) upp <- dots$upp
    if(hasArg(fun)||hasArg(cond)||!is.null(low)||!is.null(upp))
        return(var(as(x,"DiscreteDistribution"),...))
    else
        return(size(x)*prob(x)*(1-prob(x)))
    })
@
%
\item\file{skewness.R}, 
<<skew, results="asis", echo=TRUE>>=
setMethod("skewness", signature(x = "Binom"),
    function(x,  ...){
    dots <- match.call(call = sys.call(sys.parent(1)),
                       expand.dots = FALSE)$"..."
    fun <- NULL; cond <- NULL; low <- NULL; upp <- NULL
    if(hasArg(low)) low <- dots$low
    if(hasArg(upp)) upp <- dots$upp
    if(hasArg(fun)||hasArg(cond)||!is.null(low)||!is.null(upp))
       return(skewness(as(x,"DiscreteDistribution"),...))
    else
        return((1-2*prob(x))/sqrt(size(x)*prob(x)*(1-prob(x))))
    })
@
%
\item\file{kurtosis.R}, 
<<kurt, results="asis", echo=TRUE>>=
setMethod("kurtosis", signature(x = "Binom"),
    function(x,  ...){
    dots <- match.call(call = sys.call(sys.parent(1)),
                       expand.dots = FALSE)$"..."
    fun <- NULL; cond <- NULL; low <- NULL; upp <- NULL
    if(hasArg(low)) low <- dots$low
    if(hasArg(upp)) upp <- dots$upp
    if(hasArg(fun)||hasArg(cond)||!is.null(low)||!is.null(upp))
       return(kurtosis(as(x,"DiscreteDistribution"),...))
    else
        p <- prob(x)
        return((1-6*p*(1-p))/(size(x)*p*(1-p)))
    })
@
%
\end{itemize}
\end{enumerate}
The procedure will be similar for \textit{any} new class of distributions.\medskip
\begin{itemize}
  \item[Comment] In the classes in package \pkg{distr} (historically the ``oldest''
  in the development of this project), we still use \code{initialize} methods;
  this is no longer needed, if you provide generating functions; for this ``more
  recent'' approach, confer the realization of class \code{Gumbel} in package \pkg{RobExtremes}.
\end{itemize}

% -------------------------------------------------------------------------------
\section{Help needed  / collaboration welcome}
% -------------------------------------------------------------------------------
You are --- as announced on  
\href{https://distr.r-forge.r-project.org}{\tt http://distr.r-forge.r-project.org} ---
         very welcome to collaborate in this project!
See in particular
\href{https://distr.r-forge.r-project.org/HOWTO-collaborate.txt}%
{\tt https://distr.r-forge.r-project.org/HOWTO-collaborate.txt}

With this you should be able to start working.
\begin{thebibliography}{2}

\bibitem{K:R:S:04}
Ruckdeschel P. and Kohl, M. (2014):
\newblock {General Purpose Convolution Algorithm for Distributions in S4-Classes 
by means of FFT}.
\newblock {\em J. Statist. Software\/}, {\bf 59}(4): 1--25.

\bibitem{R:K:S:C:04}
Ruckdeschel P., Kohl M., Stabla T., and Camphausen F. (2006):
\newblock {S4 Classes for Distributions.} 
\newblock {\em R-News\/}, {\bf 6}(2): 10--13.
\newblock https://CRAN.R-project.org/doc/Rnews/Rnews\_2006-2.pdf
%\newblock See also {http://www.uni-bayreuth.de/departments/math/org/mathe7/RUCKDESCHEL/pubs/distr.pdf}

\end{thebibliography}
% no longer needed
%%(ll)cleanup, echo=FALSE(ggeq)
%%#unloadNamespace("SweaveListingUtils")
%%(at)
% -------------------------------------------------------------------------------
\end{document}
% -------------------------------------------------------------------------------
