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{$LastChangedDate: 2010-12-02 16:05:48 +0100 (Do, 02 Dez 2010) $}
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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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\markboth{\sl Packages ``{\tt distr}'', ``{\tt distrSim}'', ``{\tt distrTEst}'', 
``{\tt distrEx}'', ``{\tt distrTeach}'', ``{\tt distrMod}''}%
{\sl Packages ``{\tt distr}'', ``{\tt distrSim}'', ``{\tt distrTEst}'',  
``{\tt distrEx}'', ``{\tt distrTeach}'', ``{\tt distrMod}''}
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\begin{document}
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\title{{\tt S4} Classes for Distributions---a manual for packages \pkg{distr}, 
        \pkg{distrEx}, \pkg{distrMod}, \pkg{distrSim}, \pkg{distrTEst}, \pkg{distrTeach}, 
        version \pkgversion}
%,version \pkgExversion}
\author{\small Peter Ruckdeschel\thanks{Fraunhofer ITWM, Kaiserslautern}
\\[-.5ex]
\small Matthias Kohl\thanks{FH Furtwangen}
\\[-.5ex]
\small Thomas Stabla\thanks{Graf-M\"unster-Gymnasium, Bayreuth}
\\[-.5ex]
\small Florian Camphausen\thanks{West-LB, London}
\smallskip\\
\small Fraunhofer ITWM\\[-.5ex]
\small Fraunhofer Platz 1\\[-.5ex]
\small 67663 Kaiserslautern\\[-.5ex]
\small Germany\\
\small e-Mail: {\small \tt Peter.Ruckdeschel@itwm.fraunhofer.de}\medskip\\
\parbox[t]{5cm}{
\footnotesize\sffamily
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\begin{tabbing}
\footnotesize\sffamily
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 Head URL: \> \parbox[t]{6cm}{\url{\svnkw{HeadURL}}}\\[1.2ex]
 Last changed date: \> \svndate\\
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}
\maketitle
% -------------------------------------------------------------------------------
\begin{abstract}
% -------------------------------------------------------------------------------
\pkg{distr} is a package for {\sf R} from version {\tt 1.8.1} onwards that is 
distributed under {\tt LGPL-3}. Its own current version is \pkgversion.
%
The aim of this package is to provide a conceptual treatment of random variables 
(r.v.'s) by means of {\tt S4}--classes. A mother class \code{Distribution} is 
introduced with slots for a parameter and for functions {\tt r},  {\tt d}, 
{\tt p}, and {\tt q} for simulation, respectively for evaluation of density / 
c.d.f.\ and quantile function of the corresponding distribution. All 
distributions of the \pkg{stats} package are implemented as subclasses of either 
\code{AbscontDistribution} or \code{DiscreteDistribution},
which themselves are again subclasses of \code{UnivariateDistribution}. %\\
%
By means of these classes, we may automatically generate new objects of these 
classes for the laws of r.v.'s under standard mathematical univariate 
transformations and under standard bivariate arithmetical operations acting
on independent r.v.'s.
%
%From version 1.6 on, \pkg{distr} has been split up into the smaller packages
%\pkg{distr} (only distribution-classes and -methods),  \pkg{distrSim}
%(standardized treatment of simulations, also under contaminations)
%and \pkg{distrTEst} \newline(classes and methods for evaluations of statistical
%procedures on such simulations).
Package \pkg{distr} in this setting works as basic package for further extensions.
These start with package \pkg{distrEx}, covering statistical functionals like 
expectation, variance and the median evaluated at distributions, as well as
distances between distributions and basic support for multivariate and
conditional distributions. Next, from version 2.0 on, comes package \pkg{distrMod}
which uses these concepts to provide an object orientated competitor to 
\code{fitdistr} from package \pkg{MASS} in covering estimation in statistical 
models. Further on there are packages \pkg{distrSim} for the standardized 
treatment of simulations, also under contaminations and package 
\pkg{distrTEst} with classes and methods for evaluations of statistical
procedures on such simulations. Finally, from version 2.0 on, there is package
\pkg{distrTeach} to embody illustrations for basic stats courses using
our distribution classes.

%\noindent The latter two of them require package \pkg{setRNG} by 
%\href{mailto:pgilbert@bank-banque-canada.ca}{Paul Gilbert}
%to be installed from \href{http://cran.r-project.org/mirrors.html}{\tt CRAN}. 
% \\

%\noindent Additionally, mainly contributed by \cite{MK:05}, in \pkg{distrEx} we 
%extend the functionality of \pkg{distr}, providing functionals like expectation 
%or variance and distances for distributions. Also, this package contains some 
%first steps to multivariate distributions, providing classes for discrete 
%multivariate distributions and for factorized, conditional
%distributions.
% -------------------------------------------------------------------------------
\end{abstract}
% -------------------------------------------------------------------------------
\tableofcontents
\noindent
{\small Parts of this document appeared in an earlier and much shorter form in 
{\em R-News\/}, {\bf 6}(2) as {\sf ``S4 Classes for Distributions''}, 
c.f.\ \cite{R:K:S:C:04}, which in its 
published form refers to package versions 1.6, resp.\ 0.4-2. This present document 
takes into account the subsequent revisions and versions.}\medskip
% -------------------------------------------------------------------------------
\addtocounter{section}{-1}
\section{Motivation}
% -------------------------------------------------------------------------------
{\sf R} up to now contains powerful techniques for virtually
any useful distribution using the suggestive naming convention
{\tt [prefix]<name>} as functions where {\tt [prefix]} stands for
 {\tt r}, {\tt d}, {\tt p}, or {\tt q}
 and {\tt <name>} is the name of the distribution.\\
There are limitations of this concept, however:
You can only use distributions which are implemented in some library
already or for which you yourself have provided an implementation.
In many natural settings you want to formulate algorithms once for
all distributions, so you should be able to treat the actual distribution 
{\tt <name>} as sort of a variable.\\
You may of course paste together prefix and the value of {\tt <name>} as a 
string and then use \code{eval(parse(....))}. This is neither very elegant nor 
flexible, however.\\
%
Instead, we would rather like to implement the algorithm by passing an object of 
some distribution class as argument to the function. Even better though, we 
would use a generic function and let the {\tt S4}-dispatching mechanism decide 
what to do at run-time. In particular, we would like to automatically generate 
the corresponding functions {\tt r}, {\tt d}, {\tt p}, and {\tt q} for the law 
of expressions like \code{X+3Y} for objects \code{X} and \code{Y} of class 
\code{Distribution}, or, more general, of a transformation of $X$, $Y$ under a 
function $f\colon \Reals^2 \to \Reals$ which is already realized as a function 
in {\sf R}.\\
This is possible with package \pkg{distr}. As an example, try
<<echo=false,results=hide>>=
## preparation: set option withSweave to true
require(distrTEst)
require(distrEx)
require(distrTeach)
require(distrMod)
distroptions(withSweave = TRUE)
options("newDevice" = TRUE)
@
<<exam1, eval = TRUE, fig = TRUE>>=
require(distr)
N <- Norm(mean = 2, sd = 1.3)
P <- Pois(lambda = 1.2)
Z <- 2*N + 3 + P
Z
plot(Z)
p(Z)(0.4)
q(Z)(0.3)
Zs <- r(Z)(50)
Zs
@
\par
\begin{small}
\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.
\end{small}
% -------------------------------------------------------------------------------
\section{Concept}
% -------------------------------------------------------------------------------
In developing our packages, we had the following principles in mind:
We wanted to be open in our design so that our classes could easily be extended 
by any volunteer in the {\sf R} community to provide more complex classes of 
distributions as multivariate distributions, times series distributions, 
conditional distributions. As an exercise, the reader is encouraged to implement 
extrem value  distributions from the package \pkg{evd}\footnote{a solution to 
this ``homework''  may be found in the sources to \pkg{distrEx}}. The largest 
effort will in fact be the documentation\ldots\\
We also wanted to preserve naming and notation from {\sf R}-\pkg{stats}
as far as possible so that any programmer used to {\tt S} could quickly
use our package. Even more so, as the distributions already implemented to
{\sf R} are all well tested and programmed with skills we lack, we use the
existing {\tt r}, {\tt d}, {\tt p}, and {\tt q}-functions wherever possible,
only wrapping them by small code sniplets to our class hierarchy.\\
Third we wanted to use a suggestive notation for our automatically generated
methods \code{r}, \code{d}, \code{p}, and \code{q}, which we think is now 
largely achieved. All this should make intensive use of object orientation in 
order to be able to use inheritance and method overloading.
Let us briefly explain why we decided to realize \code{r}, \code{d},
\code{p}, and \code{q} as part of our class definitions:
Doing so, we place ourselves somewhere between
pure object orientation where methods would be {\it slots\/} ---in the language 
of the {\tt S4}-concept, confer \cite{Cham:98}--- and the {\tt S4} paradigm 
where methods ``live their own life'' apart from the classes, or, to \code{q}, 
which should be regarded use \cite{Beng:03}'s terminology, we use 
COOP\footnote{class-object-orientated 
programming, as e.g.\ in {\tt C++}}-style for \code{r}, \code{d}, \code{p}, and 
\code{q} methods, and FOOP\footnote{function-object-orientated programming, 
as in the {\tt S4}-concept} -style for "normal" methods.\\
The {\tt S4}-paradigm with methods which are not attached to an object but 
rather behave differently according to the classes of their arguments is fine 
if there are particular user-written methods for only some few general 
distribution classes like  \code{AbscontDistribution}, as in the case for 
\code{plot} or \code{"+"} (c.f.\ \cite{K:R:S:04}, Section 2.2). 
During a typical {\sf R} session with \pkg{distr}, however, there will be a lot 
of, mostly automatically generated objects of our distribution classes, each 
with its own \code{r}, \code{d}, \code{p}, and \code{q}; this even applies to 
intermediate expressions like \code{2*N}, \code{2*N+3} to eventually produce 
\code{Z} in the example in the motivation. Treating \code{r}, \code{d}, 
\code{p}, and \code{q} as generic functions, we would need to generate new 
classes for each expression \code{2*N}, \code{2*N+3}, \code{Z} and, 
correspondingly, particular {\tt S4}-methods for \code{r}, \code{d}, \code{p}, 
and \code{q} for each of these new classes; apparently, this would produce 
overly many classes for an effective inheritance structure. \\
In providing arithmetics for distributions, we have to deviate a little from
the paradigm of {\tt S} as a functional language: For operators like ``$+$'',
additional parameters controlling the precision of the results cannot be handily
passed as arguments. For this purpose we provide global options which may be
inspected and modified by \code{distroptions}, 
\code{getdistrOption}\footnote{Upto version 0.4-4, we used a different mechanism 
to inspect/modify global options of \pkg{distrEx} (see 
section~\ref{distrExoptions}); corresponding functions \code{distrExoptions}, 
\code{getdistrExOption} for package \pkg{distrEx} are available from version 
1.9 on.} in complete analogy to \code{options}, \code{getOption}.
%
Finally our concept as to parameters: Contrary to the standard {\sf R}-functions 
like \code{rnorm} we only permit length $1$ for parameters like \code{mean}, 
because we see the objects as implementations of univariate random variables, 
for which vector-valued parameters make no sense; rather one could gather 
several objects with possibly different parameters to a vector/list of 
distributions. Of course, the original functions \code{rnorm} etc.\ remain 
unchanged and still allow for vector-valued parameters.
Kouros Owzar in an off-list mail raised the point, that in case of multiple 
parameters as in case of the normal or the $\Gamma$-distribution, it might be 
useful to be able to pass these multiple parameters in vectorized form to the 
generating function. We, too, think that this is a good idea, but have 
shifted this question to the new extension package \pkg{distrMod} which covers
more general treatment of statistical models, see section~\ref{distrMod}.
% -------------------------------------------------------------------------------
\section{Organization in classes}
% -------------------------------------------------------------------------------
Loosely speaking we have three large groups of classes: distribution classes (in 
\pkg{distr}), simulation classes (in \pkg{distrSim}) and an evaluation class (in 
\pkg{distrTEst}), where the latter two are to be considered only as tools which 
allow a unified treatment of simulations and evaluation of statistical estimation 
(perhaps also tests and predictions later) under varying simulation situations.
Additionally, package \pkg{distrEx} provides classes for discrete multivariate 
distributions and for factorized, conditional distributions, as well as a bundle 
of functionals and distances (see below).
% -------------------------------------------------------------------------------
\subsection{Distribution classes}
% -------------------------------------------------------------------------------
The purpose of the classes derived from the class \code{Distribution}  is to 
implement the concept of a r.v./distribution as such in {\sf R}.\\
All classes derived from \code{Distribution} have a slot \code{param} for a 
parameter, a slot \code{img} for the range and the constitutive slots \code{r}, 
\code{d}, \code{p}, and \code{q}.\\
From version 1.9 on, up to arguments referring to a parameter of the 
distribution (like \code{mean} for the normal distribution), these function 
slots have the same arguments as those of package \pkg{stats}, i.e.; for a 
distribution object \code{X} we may call these functions as 

\begin{itemize}
\item \code{r(X)(n)}  $\qquad$ ---except for objects of class \code{Hyper},
where there is a slot \code{n} already, so here the argument name 
to \code{r} is \code{nn}.
\item \code{d(X)(x, log = FALSE)}
\item \code{p(X)(q, lower.tail = TRUE, log.p = FALSE)}
\item \code{q(X)(p, lower.tail = TRUE, log.p = FALSE)}
\end{itemize}

For the arguments of these function slots see e.g.\ \code{rnorm}
from package \pkg{stats}.
Note that, as usual, slots \code{d}, \code{p}, and \code{q} are vectorized
in their first argument, but are not on the subsequent ones.
The idea is to gain higher precision for the upper tails or when multiplying
probabilities.
% -------------------------------------------------------------------------------
\subsubsection{Subclasses}
% -------------------------------------------------------------------------------
To begin with, we have considered univariate distributions giving the 
{\tt S4}-class \code{UnivariateDistribution}, and as typical subclasses, we 
have introduced classes for absolutely continuous and discrete distributions 
---\code{AbscontDistribution} and \code{DiscreteDistribution}.\\

The former, from version 1.9 on, has a slot \code{gaps} of class 
\code{OptionalMatrix}, i.e.; an object which may either be \code{NULL} or
a \code{matrix}. This slot, if non-\code{NULL}, contains left and right 
endpoints of intervals where the density of the object is $0$. This slot 
may be inspected by the accessor \code{gaps()} and modified by a corresponding
replacement method. It may also be filled automatically by 
\code{setgaps(object, exactq = 6, ngrid = 50000)}, where upon evaluation of 
the \code{d}-slot on a grid of length \code{ngrid}, all regions in the 
range\footnote{more precisely: between lower and upper \code{TruncQuantile};
 \code{TruncQuantile} is a global option of  \pkg{distr} described in 
 section~\ref{options}} of the distribution where the density is smaller than 
 $10^{\scriptscriptstyle - {\rm exactq}}$ are set to gaps.
 Internally, we have helper functions \code{.consolidategaps} to merge adjacent
 intervals and \code{mergegaps} to merge \code{slots} of different objects.
 \\ For saved objects 
 from earlier versions, we provide the functions \code{isOldVersion} and 
 \linebreak[4]\code{conv2NewVersion} to check whether the
object was generated by an older version of this package and
to convert such an object to the new format, respectively.\\

Class \code{DiscreteDistribution} has a slot \code{support}, a vector containing 
the support of the distribution, which is truncated to the lower/upper 
\code{TruncQuantile} in case of an infinite support. \code{TruncQuantile} is a 
global option of  \pkg{distr} described in section~{\ref{options}}. 


Also from version 1.9 on, class \code{DiscreteDistribution} has a subclass 
\code{LatticeDistribution} for supports consisting of\footnote{or at least
if filled with points carrying no mass have a representation as an affine linear 
lattice} an affine linear lattice of form $p+iw$ for $p\in\R$, $w\in\R$, 
$w\not=0$ and $i=0,1,\ldots,L$, 
$L\in\N \cup\infty$. This class gains a slot \code{lattice} of 
class \code{Lattice} (see below). The purpose of this class is mainly its use 
in DFT/FFT methods for convolution. Slot \code{lattice} may be 
inspected by the usual accessor function \code{lattice()}.
As by inheritance, all subclasses of \code{LatticeDistribution} which prior to
version 1.9 were direct subclasses of \code{DiscreteDistribution} gain a 
slot \code{lattice}, too, we provide again \code{isOldVersion} and 
\code{conv2NewVersion} methods to check whether the object was generated by an 
older version of this package and to convert such an object to the new 
format, respectively. Also note that internally, we suppress lattice points from 
the support where the probability is $0$.\\        


Objects of classes \code{LatticeDistribution} resp.\ 
\code{DiscreteDistribution}, and from version 2.0 on, also 
\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, fig = TRUE>>=
D <- DiscreteDistribution(supp = c(1,5,7,21), prob = c(0.1,0.1,0.6,0.2))
D
plot(D)
@                 
\newline
and to generate an absolutely continuos distribution with density proportional
to $e^{-|x|^3}$, we write
<<AbscDist, eval = TRUE, fig = TRUE>>=
AC <- AbscontDistribution(d = function(x) exp(-abs(x)^3), withStand = TRUE)
AC
plot(AC)
@                 
\newline
As subclasses of these absolutely continuous and discrete classes, we have 
implemented all parametric families which already exist in the  \pkg{stats} 
package of {\sf R} in form of 
{\tt [prefix]<name>} functions ---by just providing wrappers to the original 
{\sf R}-functions.\\
%
Schematically, the inheritance relations as well as the slots 
of  the corresponding classes may be read off from Figure~\ref{fig1c}. 
Class \code{LatticeDistribution} and slot \code{gaps}, as well as
additional classes \code{AffLinAbscontDistribution}, 
\code{AffLinDiscreteDistribution}, \code{AffLinLatticeDistribution} 
(c.f.\ section~\ref{afflin}) are still lacking in this graphic so far, however,
as well as the classes introduced in version 2.0.
\\

\ifpdf
\begin{figure}[!ht]\label{fig1}
\vspace{2ex}
  \begin{center}
%    \includegraphics[viewport=0 0 500 700,width=9cm]{distribution.pdf}%
    \includegraphics[viewport=130 150 500 750,width=9cm]{distribution.pdf}%
    \caption{\label{fig1c}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{Distribution} where we do not
    repeat inherited slots
    }}
  \end{center}
\vspace{-4ex}
\end{figure}
\else
\begin{figure}[htb]\label{fig1}
  \begin{center}
    \includegraphics[viewport=130 150 500 730,width=7.5cm]{distribution.ps}%
    \caption{\label{fig1c}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{Distribution} where we do not 
    repeat inherited slots
    }}
  \end{center}
\vspace{-1ex}
\end{figure}
\fi

The most powerful use of our package probably consists in operations to 
automatically generate new slots \code{r}, \code{d}, \code{p}, and \code{q} 
---induced by mathematical transformations. This is discussed in some detail in 
subsection~{\ref{methods}}.

\subsubsection{Classes for Mixture Distributions}

\paragraph{Lists of distributions}

As a first step, we allow distributions to be gathered in lists, giving
classes \code{DistrList} and \code{UnivarDistrList}, where in case of the latter,
all elements must be univariate distributions. For these, the usual indexing 
operations with \code{[[.]]} are available. As we will use these lists to 
construct more general mixture distributions in some subsequent versions, we 
have moved these routines to package \pkg{distr} from version 1.9 on.


\paragraph{Mixing distributions}

To be able to work with distributions which are neither purely absolutely continuous
nor purely discrete, like e.g.\ the distribution of $\min(X,1)$ for 
$X\sim{\cal N}(0,1)$, from package version 2.0 on, we support mixtures of 
distributions. These are realized as subclasses of class \code{UnivariateDistribution}.
To begin with, we introduce a class \code{UnivarMixingDistribution} as subclass of
class \code{UnivariateDistribution} which additionally has two slots \code{MixCoeff}
and \code{MixDistr}. While the former is a numeric vector taking up the mixture
coefficients of the distribution, the latter is an object of class 
\code{UnivarDistrList} as described below, taking up the distributions of the
mixture components; as usual, these slots have their respective accessor and
replacement functions. Usually, this mixing distribution will neither have 
a Lebesgue density nor be purely discrete, having a counting density. So slot
\code{d} as a rule will be empty. Objects of this class may be generated by
the generating function \code{UnivarMixingDistribution()},  see also
the corresponding help. In addition there is the function \code{flat.mix}
to simplify such an object converting it to an object of class 
\code{UnivarLebDecDistribution}; confer subsection~\ref{flat}.
Note that these mixing distributions may be recursive, i.e.\ compoments of 
slot \code{MixDistr} may again
be of class \code{UnivarMixingDistribution}.
<<examLis, eval = TRUE>>=
library(distr)
M1 <- UnivarMixingDistribution(Norm(), Pois(lambda=1), Norm(), 
      withSimplify = FALSE)
M2 <- UnivarMixingDistribution(M1, Norm(), M1, Norm(), withSimplify = FALSE)
M2
@

\paragraph{Lebesgue Decomposed distributions}

As seen in the above example of $\min(X,1)$, classes \code{DiscreteDistribution}
and \code{Abscontdistribution} are not closed under arithmetic operations. To
have such a closure, from version 2.0 on, we introduce class
\code{UnivarLebDecDistribution}, which realizes a Lebesgue decomposition of a 
univariate distribution into a discrete and an absolutely continuous distribution.
Of course, we still cannot cover distributions having a non-trivial continuous
but not absolutely continuous part like the Cantor distribution, but class 
\code{UnivarLebDecDistribution} provides a sufficiently general compromise.
Class \code{UnivarLebDecDistribution} is a subclass of class 
\code{UnivarMixingDistribution}, where in addition we assume that both slots
\code{MixCoeff} and \code{MixDistr} are of length 2, and that the first component
of slot  \code{MixDistr} is of class \code{AbscontDistribution} while the second
is of class \code{DiscreteDistribution}. For this class there are particular
accessors \code{acWeight}, \code{discreteWeight} for the respective weights and
\code{acPart}, \code{discretePart} for the respective distributions. Again there
is a generating function \code{UnivarMixingDistribution()}.
In addition there is the function \code{flat.LCD}
to simplify such an object converting it to an object of class 
\code{UnivarLebDecDistribution}; confer subsection~\ref{flat}.%
Classes \code{AbscontDistribution}, \code{DiscreteDistribution} and
\code{UnivarLebDecDistribution}  are grouped to a virtual class 
(more specifically a class union)
\code{AcDcLcDistribution}.

\paragraph{Compound distributions}
From version 2.1 on, we also support compound distributions, i.e. the distributions
$D$ of form $D={\cal L}(\sum_{i=1}^N X_i)$, $X_i\stackrel{\rm\scriptsize i.i.d.}{\sim} F$,
a distribution on $\R$, and, independent of the $X_i$, $N\sim Q$ a distribution on $\N_0$.
These distributions are implemented as class \code{CompoundDistribution} which is a subclass
of class \code{UnivarMixingDistribution}; in addition this class has 
two slots, \code{NumbOfSummandsDistr}, the distribution $Q$ of the number of
summands (or frequency distribution) and \code{SummandsDistr} the distribution $F$
of the summands. Correspondingly, in package \pkg{distrEx} there are specialized
expectation and variance methods.

\subsubsection{Classes for multivariate distributions and for conditional 
distributions}

In \pkg{distrEx}, besides providing particular distribution classes
for the Gumbel distribution (see \code{?Gumbel}) and the Pareto 
distribution (see \code{?Pareto}), 
we provide the following classes for handling multivariate 
distributions:


\paragraph{Multivariate distribution classes}

Multivariate distributions are much more complicated than univariate ones,
which is why but a few exceptional ones have already been implemented to R in
packages like \pkg{multnorm}. In particular it is not so clear what a slot 
\code{q} should mean and, in higher dimensions slot \code{p}, and possibly also 
slot \code{d} may become awkward. So, for multivariate distributions, realized 
as class \code{MultivariateDistribution}, we only insist on slot \code{r}, while 
the other functional slots may be left void.

The easiest case is the case of a discrete multivariate distribution with finite 
support which is implemented as class \code{DiscreteMVDistribution}.

\paragraph{Conditional distribution classes}

Also arising in multivariate settings only are conditional distributions. In our 
approach, we realize factorized, conditional distributions where the 
(factorized) condition is in fact treated as an additional parameter to the 
distribution. The condition is realized as an object of class \code{Condition}, 
which is a slot of corresponding classes \code{UnivariateCondDistribution}. 
This latter is the mother class to classes
\code{AbscontCondDistribution} and \code{DiscreteCondDistribution}.
The most important application of these classes so far is regression, where
the distribution of the observation given the covariates is just realized as
a \code{UnivariateCondDistribution}.

\subsubsection{Parameter classes}
%
As most distributions come with a parameter which often is of own interest, we 
endow the corresponding slots of a distribution class with an own parameter 
class, which allows for some checking like ``Is the parameter \code{lambda} of 
an exponential distribution non-negative?'',
``Is the parameter \code{size} of a binomial a positive integer?''\\
Consequently, we have a method \code{liesIn} that may answer such questions by a 
\code{TRUE}/\code{FALSE} statement. Schematically, the inheritance relations of 
class \code{Parameter} as well as the slots of the corresponding
\mbox{(sub-)}classes may be read off in Figure~\ref{fig4c} where we do not 
repeat inherited slots.
%
\ifpdf
\begin{figure}[!ht]\label{fig4}
  \begin{center}
% \includegraphics[viewport=30 10 540 390,height=11cm,width=12cm]{parameter.pdf}
% [viewport=30 10 540 390,width=12.cm]
  \includegraphics[viewport=30 80 400 690,height=15cm,width=12cm]{parameter.pdf}
    %[viewport=30 10 540 390,width=12.cm]
    \caption{\label{fig4c}{\footnotesize Inheritance relations and slots of the 
     corresponding \mbox{(sub-)}classes for \code{Parameter}
    }}
  \end{center}
\end{figure}
\else
\begin{figure}[htb]\label{fig4}
  \begin{center}
    \includegraphics[viewport=80 120 340 600,height=12.8cm,width=9.0cm, 
                     angle=-90]{parameter.ps}%
    \caption{\label{fig4c}{\footnotesize Inheritance relations and slots of the 
              corresponding \mbox{(sub-)}classes
    for \code{Parameter}
    }}
  \end{center}
\end{figure}
\fi
%
The most important set to be used as parameter domain/sample space
 (\code{rSpace}) will be an Euclidean space. So \code{rSpace} and 
 \code{EuclideanSpace} are also implemented as classes,
 the structure of which may be read off in Figure~\ref{fig5c}.
\begin{figure}[htb]\label{fig5}
  \begin{center}
    \ifpdf
    \includegraphics[%viewport=0 30 600 600,
    width=3.9cm]{rspace.pdf}
    \else
    \includegraphics[width=3.7cm]{rspace.ps}%
    \fi
    \caption{\label{fig5c}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes
    for \code{rSpace}
    }}
  \end{center}
\end{figure}

From version 1.9 on, we also have a subclass \code{Lattice}, which is still 
lacking in the preceding figure. It has slots \code{pivot} (of class 
"numeric"), \code{width} (of class "numeric" but tested against ``{\tt ==0}'')
and  \code{Length} (of class "numeric" but tested to be an integer 
``{\tt >0}'' or {\tt Inf}). 
All slots may be inspected/modified by the usual accessor/replacement functions.

\subsection{Simulation classes}
From version 1.6 on, the classes and methods of this subsection are available in 
package  \pkg{distrSim}.

The aim of simulation classes is to gather all relevant information about a 
simulation in a correspondingly designed class. To this end we introduce the 
class \code{Dataclass} that serves as a common mother class for both "real" and 
simulated data.
%
As derived classes we then have a simulation class where we also gather all 
information needed to reconstruct any particular simulation.\\
%
%\newline0??????\\
From version 1.8 of this package on, we have changed the
format how data / simulations are stored:
In order to be able to cope with multivariate,
regression  and (later) time series distributions,
we have switched to the common array format
%
{\tt samplesize x obsDim x runs} where {\tt obsDim} is the dimension of the 
observations.
%
For saved objects from earlier versions, we provide the functions
\code{isOldVersion} and \code{conv2NewVersion} to check whether the
object was generated by an older version of this package and
to convert such an object to the new format, respectively.
For objects generated from version 1.8 on, you get the package 
version of package~\pkg{distrSim}, under which they have been generated
by a call to \code{getVersion()}. \\
%
%??????1\\
 Finally, coming from robust statistics
we also consider situations where the majority of the data stems from an ideal 
situation/distribution whereas a minority comes from a contaminating source. To 
be able to identify ideal and contaminating observations, we also store this 
information in an indicator variable.\\
%
As the actual values of the simulations only play a secondary role, and as the 
number of simulated variables can become very large, but still easily 
reproducible, it is not worth storing all simulated observations but rather only 
the information needed to reproduce the simulation.
This can be done by \code{savedata}.\\
Schematically, the inheritance relations of class \code{Dataclass} as well as 
the slots of the corresponding \mbox{(sub-)}classes may be read off in 
Figure~\ref{fig2c} where we do not  repeat inherited slots.
\begin{figure}[htb]\label{fig2}
  \begin{center}
    \ifpdf
    \includegraphics[viewport=120 30 470 250,width=8.5cm]{dataclass.pdf}
    \else
    \includegraphics[width=12.5cm]{dataclass.ps}%
    \fi
    \caption{\label{fig2c}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes
    for \code{Dataclass}
    }}
  \end{center}
\end{figure}
%\newline0??????\\
Also, analogously to  package %s
 \pkg{distr}, %and \pkg{distrEx},
 global options for the output by methods \code{plot} and \code{summary}
are controlled by \code{distrSimoptions()} and \code{getdistrSimoptions()}
\smallskip\\
%??????1\\

\subsection{Evaluation class}
From version 1.6 on, the class and methods of
this subsection are available in package  \pkg{distrTEst}. \\
When investigating properties of a new procedure (e.g. an estimator) by means of 
simulations, one typically evaluates this procedure on a large set of simulation 
runs and gets a result for each run. These results are typically not available 
within seconds, so that it is worth storing them.
To organize all relevant information about these results, we introduce a class 
\code{Evaluation} the slots of which is filled by method \code{evaluate} ---see 
subsection~\ref{evaluate}. Schematically, the slots of this class may be read 
off in Figure~\ref{fig3c}.
\begin{figure}[htb]\label{fig3}
  \begin{center}
    \ifpdf
    \includegraphics[viewport=60 20 420 200,width=5cm]{evaluation.pdf}
    \else
    \includegraphics[width=5cm]{evaluation.ps}%
    \fi
    \caption{\label{fig3c}{\footnotesize Slots of class \code{Evaluation}}}
  \end{center}
\end{figure}
%\newline0??????\\
A corresponding \code{savedata} method
saves the object of class \code{Evaluation} in two files in the {\sf R}-working 
directory: one using the filename \code{<filename>} also stores the results; the 
other one, designed to be ``human readable'', comes as a comment file with 
filename \code{<filename>.comment}
only stores the remaining information.
The filename can be specified in the optional argument \code{fileN} to 
\code{savedata}; by default it is concatenated from the \code{filename} slot of 
the \code{Dataclass} object and \code{<estimatorname>}, which you may either
pass as argument \code{estimatorName} or by default is taken as the {\sf R}-name 
of the corresponding {\sf R}-function specified in slot \code{estimator}.

From version 1.8 on, slot \code{result} in class \code{Evaluation} is of class 
\code{DataframeorNULL}, i.e.; may be either a data frame or {\tt NULL}, and slot 
\code{call.ev} in class \code{Evaluation} is of class "CallorNULL", i.e.; may be 
either a call or {\tt NULL}. Also, we want to gather \code{Evaluation} objects 
in a particular data structure \code{EvaluationList}
(see below), so we have to be able to check whether all data sets in the 
gathered objects coincide.
For this purpose, from this version on, class \code{Evaluation} has an 
additional slot \code{Data} of class \code{Dataclass}. In order not to burden 
the objects of this class too heavily with uninformative
simulated data, in case of a slot \code{Data} of one of the simulation-type 
subclasses of \code{Dataclass},
this \code{Data}  itself has an empty \code{Data}-slot.\\

\subsection{EvaluationList class}
The class and methods of this subsection are available in package 
 \pkg{distrTEst}. \\
In order to compare different procedures / estimators for the
same problem, it is natural to gather several \code{Evaluation} objects
with results of the same range (e.g.\ a parameter space) generated on the
same data, i.e.; on the same \code{Dataclass} object. To this end, from
version 1.8 on, we have introduced class \code{EvaluationList}.
Schematically, the slots of this class may be read off in Figure~\ref{fig3c1}.
\begin{figure}[htb]\label{fig3-1}
  \begin{center}
    \ifpdf
    \includegraphics[viewport=0 0 436 185,width=5.5cm]{EvaluationList.pdf}%
    \else
    \includegraphics[viewport=0 0 436 185,width=5.5cm]{EvaluationList.ps}%
    \fi
%\parbox{10cm}{TO BE DONE}
    \caption{\label{fig3c1}{\footnotesize Slots of class \code{EvaluationList}}}
  \end{center}
\end{figure}
The common \code{Data} slot of the \code{Evaluation} objects in an 
\code{EvaluationList} object may be accessed by the accessor method \code{Data}.
%??????1\\
%
%
\section{Methods}\label{methods}
%
\subsection{Arithmetics}
We have made available quite general arithmetical operations to our distribution 
objects, generating new image distributions automatically. In this context some
comments are due as to the interpretation of corresponding arithmetic 
expressions of distribution objects: 

\begin{description}
  \item[{\Large \sc Caveat:\/}] {\bf These arithmetics
operate on the corresponding r.v.'s and {\bf not} on the distributions.}
\end{description}
(For the latter, they only would make sense in restricted cases like convex 
 combinations).\\

Martin M\"achler pointed out that this might be confusing. So, this warning is 
also issued on attaching package \pkg{distr}, and,  by default, again whenever a 
\code{Distribution} object, produced by such arithmetics is shown or printed; 
this also applies to the last line in
<<warningArith, eval = TRUE>>=
  A1 <- Norm(); A2 <- Unif()
  A1 + A2
@
\begin{verbatim}
Warning message:
arithmetics on distributions are understood as operations on r.v.'s
see 'distrARITH()'; for switching off this warning see '?distroptions' in: 
print(object)
\end{verbatim}
This behaviour will soon be annoying so you may switch it off setting the global 
option \code{WarningArith} to \code{FALSE} (see section~\ref{options}).\medskip\\
%
Function \code{distrARITH()} displays the following comment
\begin{footnotesize} 
\begin{quote} 
\begin{verbatim}
######################################################################
#  On arithmetics operating on distributions in package "distr"       
######################################################################


                      Attention:

Special caution is due in the followin issues

%--------------------------------------------------------------------
% Interpretation of arithmetics
%--------------------------------------------------------------------

Arithmetics on distribution objects are understood as operations on 
corresponding random variables (r.v.'s) and _not_ on distribution 
functions or densities; 
e.g.

    sin( Norm() + 3 * Norm() ) + 2
    
returns a distribution object representing the distribution of the r.v. 

     sin(X+3*Y)+2
     
where X and Y are r.v.'s i.i.d. N(0,1).

%--------------------------------------------------------------------
% Adjusting accuracy
%--------------------------------------------------------------------

Binary operators like "+", "-" would loose their elegant calling
 e1 + e2 if they had to be called with an extra argument controlling
their accuracy. Therefore, this accuracy is controlled by global options.
These options are inspected and set by distroptions(), getdistrOption(),
see ?distroptions

%--------------------------------------------------------------------
% Multiple instances in expressions and independence
%--------------------------------------------------------------------

Special attention has to be paid to arithmetic expressions of
distributions involving multiple instances of the same symbol:

/-> All arising instances of distribution objects in arithmetic
    expressions are assumed stochastically independent. <-/

As a consequence, whenever in an expression, the same symbol for 
an object occurs more than once, every instance means a new 
independent distribution.  

So for a distribution object X, the expressions X+X and 2*X are _not_
equivalent. 

The first means the convolution of distribution X with distribution 
X, i.e. the distribution of the r.v. X1 + X2, where X1 and X2 are 
identically distributed according to X. 
In contrast to this, the second expression means the distribution of 
the r.v. 2 X1 = X1 + X1, where again X1 is distributed according to X.

Hence always use 2*X, when you want to realize the second case.

Similar caution is due for X^2 and X*X and so on.

%--------------------------------------------------------------------
% Simulation based results varying from call to call
%--------------------------------------------------------------------

At several instances (in particular for non-monotone functions from group
 Math like sin(), cos()) new distributions are generated by means of 
RtoDPQ, RtoDPQ.d, RtoDPQ.LC. In these functions, slots d, p, q are 
filled by simulating a large number of random variables, hence they are 
stochastic estimates. 

So don't be surprised if they will change from call to call.
\end{verbatim}
\end{quote} 
\end{footnotesize}
%
\subsection{Affine linear transformations}\label{afflin}
%
We have overloaded the operators \code{"+"}, \code{"-"}, \code{"*"}, \code{"/"} 
such that affine linear transformations which involve only single univariate 
r.v.'s are available;
i.e.\ is expressions like \code{Y=(3*X+5)/4} are permitted for an object 
\code{X} of class \code{AbscontDistribution}
or \code{DiscreteDistribution}
(or some subclass), giving again an object \code{Y} of
class \code{AbscontDistribution} or \linebreak[4]\code{DiscreteDistribution} 
(in general). 
Here the corresponding transformations of the \code{d}, \code{p}, and 
\code{q}-functions are done analytically.\\
%
From version 1.9 on, we use subclasses 
\code{AffLinAbscontDistribution}, \code{AffLinDiscreteDistribution},
\code{AffLinLatticeDistribution} as classes of the return values to enhance
accuracy of functinals like \code{E}, \code{var}, etc.\ 
in package \pkg{distrEx}. These classes in addition
to their counterparts without prefix ``\code{AffLin}'' have slots \code{a}, \code{b},
and \code{X0}, to capture the fact that an object of this class is distributed
as \code{a * X0 + b}. Also, we introduce a class union \code{AffLinDistribution}
of classes \code{AffLinAbscontDistribution} and
\linebreak[4]\code{AffLinDiscreteDistribution}.
Consequently, the result \code{Y} of \code{Y <- a1 * X + b1} for an 
object \code{X} of (a subclass of) class \code{AffLinDiscreteDistribution}
(if \code{ a != 0}) is of the same class as \code{X} but with slots
\code{Y@a = a1 * X@a}, \code{Y@b = b1 + a1 * X@b}, \code{Y@X0 = X@X0}.
In version 2.0, the same principle has been applied to introduce class
\code{AffLinUnivarLebDecDistribution}. All \code{AffLin}-xxx distribution
classes are grouped to a virtual class (more specifically a class union)
\code{AffLinDistribution}.
%
\subsection{Decompositions, Flattening and Other Simplifications}\label{flat}
%
\paragraph{Decompositions: }
One of the issues when programming the distribution of the multiplication of
independent random variables is that we have to treat positive and negative
part (and, if nontrivial, point mass to $0$) separately. To this end, from
version 2.0 on, there are methods \code{decomposePM} to decompose a
discrete, an absolutely continuous or a Lebesgue decomposed distribution into
its respective parts.  
%
<<examdcP, eval = TRUE>>=
decomposePM(Norm())
     decomposePM(Binom(2,0.3)-Binom(5,.4))
     decomposePM(UnivarLebDecDistribution(Norm(),Binom(2,0.3)-Binom(5,.4), 
                 acWeight = 0.3))
@                 

\paragraph{Simplification by flattening: }
On the other hand, concatenating mathematical operations would easily
yield quite complicated structures. A first thing to do is to look whether
some components carry mass (approximately) 0. \code{simplifyD} uses this to
cancel out such components, and if possible return simpler types; see also
the help to this function.

Also, sometimes one would like to let collapse a whole list of distributions
(as in the \code{MixDistr} of a \code{UnivarMixingDistribution} object) 
into a simpler \code{UnivarLebDecDistribution}-class
form. This is what is done in the the functions \code{flat.mix} and
\code{flat.LCD}. 

<<examflat, eval = TRUE>>=
D1 <- Norm()
D2 <- Pois(1)
D3 <- Binom(1,.4)
D4 <- UnivarMixingDistribution(D1,D2,D3, mixCoeff = c(0.4,0.5,0.1), 
      withSimplify = FALSE)
D <- UnivarMixingDistribution(D1,D4,D1,D2, mixCoeff = c(0.4,0.3,0.1,0.2), 
      withSimplify = FALSE)
D
D0<-flat.mix(D)
D0
@                 

Many arithmetic operations described in the subsequent sections do
this simplification on their return value, according to the global option
\code{SimplifyD}.

\paragraph{Simplification by collapsing: }
Dealing with discrete distributions, several arithmetical/mathematical
operations tend to create new quite unequally spaced supports (mostly by joining supports
of operands), often accumulating support points somewhere. In order to overcome
this, from version 2.1 on, following a proposal by  
\href{mailto:jacobvanetten@yahoo.com}{Jacob van Etten},
whenever support points get closer to each other than prescribed in 
global option \code{DistrResolution} (see also section~\ref{options}),
and if in addition global option \code{DistrCollapse} is \code{TRUE},
we collapse these support point, using the median of the collapsed points
as new support point, to which we attribute the cumulated probability mass. 
If in addition global option \code{DistrCollapse.Unique.Warn} is \code{TRUE}
we issue a warning on collapsing occasions.

\subsection[The group math of unary mathematical operations]{The group 
\code{math} of unary mathematical operations}
%
Also the group \code{math} of unary mathematical operations is available for
distribution classes; so
expressions like \code{exp(sin(3*X+5)/4)} are permitted.
%
 The corresponding \code{r} method consists in simply
performing the transformation to the simulated values of \code{X}.
The corresponding (default-) \code{d}, \code{p} and \code{q}-functions are 
obtained by simulation, using the technique described in the following 
subsection.\\
By means of \code{substitute}, the bodies of the \code{r}, \code{d}, 
\code{p}, \code{q}-slots of distributions show the parameter values with 
which they were generated; in particular,
convolutions and applications of the group \code{math} may be traced in
the \code{r}-slot of a distribution object, compare\newline
\code{r(sin(Norm()) + cos(Unif() * 3 + 2))}.

Initially, it might be irritating that the same ``arithmetic'' expression
evaluated twice in a row gives two different results, compare
<<arithmetic, eval = TRUE>>=
  A1 <- Norm(); A2 <- Unif()
  d(sin(A1 + A2))(0.1)
  d(sin(A1 + A2))(0.1)
  sin(A1 + A2)
@
This is due to the fact, that all slots are filled starting from simulations.
To explain this, a warning is issued  by default, whenever a \code{Distribution}
object, filled by such simulations is shown or printed; this also applies to the 
last line in the preceding code sniplet. This behaviour may again be switched 
off by setting the global option \code{WarningSim} to \code{FALSE} 
(see section~\ref{options}).\\

As they are frequently needed, from version 1.9 on, math operations 
\code{abs()}, \code{exp()}, and ---if an {\sf R}-version $\ge$ {\tt 2.6.0} is 
used--- also \code{log()} are implemented in an analytically exact form, 
i.e.; with exact expressions for slots \code{d}, \code{p}, and \code{q}.

%
\subsection{Construction of \code{d}, \code{p}, and \code{q} from \code{r}}
%
In order to facilitate automatic generation of new distributions, in particular 
those arising as image distributions under transformations of correspondingly 
distributed random variables, we provide ad hoc methods that should be 
overloaded by more exact ones wherever possible. As, at least in principle
each of these slots is sufficient for the reconstruction of the other ones,
we follow the following strategy:
\\
\begin{center}
\begin{tabular}{llll|p{10cm}}
\Code{d} & \Code{p} &  \Code{q} & \Code{r} & reconstruction\\
\hline
$+$&$+$&$+$&$+$&no reconstruction necessary\\
$+$&$+$&$+$&$-$&\Code{r} as \Code{q(X)(runif(n))}\\
$+$&$+$&$-$&$+$&\Code{q} by numerical inversion from \Code{p}\\
$+$&$+$&$-$&$-$&\Code{q} again from \Code{p} and 
                         \Code{r} again from slot \Code{q}\\
$+$&$-$&$+$&$+$&\Code{p} by numerical integration from \Code{d}\\
$+$&$-$&$+$&$-$&\Code{p} from \Code{d}, and \Code{r} from  \Code{q}\\
$+$&$-$&$-$&$+$&\Code{p} from \Code{d}, and \Code{q} from  \Code{p}\\
$+$&$-$&$-$&$-$&\Code{p} from \Code{d}, 
\Code{q} from \Code{p} and \Code{r} from  \Code{q}\\
$-$&$+$&$+$&$+$&\Code{d} by numerical differentiation (with \Code{D1ss}
from package \pkg{sfsmisc} from \Code{p}\\
$-$&$+$&$+$&$-$&\Code{d} from \Code{p}, \Code{r} from \Code{q}\\
$-$&$+$&$-$&$+$&\Code{d}, \Code{q}  from \Code{p}\\
$-$&$+$&$-$&$-$&\Code{d}, \Code{q}  from \Code{p}, \Code{r} from \Code{q}\\
$-$&$-$&$+$&$+$&\Code{p} by numerical inversion from \Code{q},
                \Code{d} from \Code{p}\\
$-$&$-$&$+$&$-$&\Code{p}, \Code{r} from \Code{q}, \Code{d} from \Code{p}\\
$-$&$-$&$-$&$+$& use \Code{RtoDPQ}\\
$-$&$-$&$-$&$-$&not allowed\\
\end{tabular}\\
\end{center}
More specifically, by means of the function 
\code{RtoDPQ} we first generate $10^{\footnotesize\tt RtoDPQ.e}$
random numbers where \code{RtoDPQ.e} is a global option of this package and is 
discussed in section~{\ref{options}}. %
A density estimator is evaluated along this sample, the distribution function is 
estimated by the empirical c.d.f. and, finally, the quantile function is 
produced by numerical inversion.
Of course the result is rather crude as it relies on the law of large numbers 
only, but this way all transformations within the group \code{math} become 
available.
If the input of the transformation is of class \code{UnivarLebDecDistribution},
\code{RtoDPQ} is replaced by \code{RtoDPQ.LC}. In this case, replicated values
are taken as belonging to the discrete part, for which the distribution is 
generated according to the corresponding frequencies with the generating
function \code{DiscreteDistribution()}. With the remaining, non replicated
values, the absolutely continuous part is reconstructed just as with \code{RtoDPQ}.

Where laws under transformations can easily be computed exactly ---as for affine
linear transformations--- we replace this procedure by the exactly transformed
\code{d}, \code{p}, \code{q}-methods.
%
\subsection{Convolution}
%
A convolution method for two independent r.v.'s is implemented by means 
of explicit calculations for discrete summands, and by means of 
DFT/FFT\footnote{Details to be found in \cite{K:R:S:04}} if one of the summands is 
absolutely continuous or (from version 1.9 on:) both are lattice distributed 
with a common lattice as support.
This method automatically generates the law of the sum of two independent 
variables/distributions $X$ and $Y$ of any univariate distributions ---or 
in {\tt S4}-jargon: the addition operator \code{"+"} is overloaded for two 
objects of class \code{UnivariateDistribution} and corresponding subclasses.
%
\subsection{Further Binary Operators}
%
Having implemented a class for Lebesgue decomposed distributions, we have been
able to realize further binary operators, in particular we have exact
analytical constructions for multiplication, division, exponentiation:
<<arith2v1, eval = TRUE, fig = TRUE>>=
  A1 <- Norm(); A2 <- Unif()
  A1A2 <- A1*A2
  plot(A1A2)
@
<<arith2v2, eval = TRUE, fig = TRUE>>=
  A12 <- 1/(A2 + .3)
  plot(A12) 
@
<<arith2v3, eval = TRUE, fig = TRUE>>=
  B <- Binom(5,.2)+1
  A1B <- A1^B
  plot(A1B, xlim=c(-3,3))
@
<<arith2V4, fig = TRUE>>=
  plot(1.2^A1)
@
<<arith2v5, fig = TRUE>>=
  plot(B^A1)
@
%
\subsection{Truncation, Pairwise Minimum/Maximum, Huberization}\label{TruncMin}
%
Up to version 2.0, we have had truncation, Huberization and minimum
and maximum of random variables as illustrating demos; in particular
the last three could not be realized in a completely satisfactory manour,
as Lebesgue decomposed distributions had not been available before.
Now these illustrations have moved into the package itself:
<<Hub, eval = TRUE, fig = TRUE>>=
H <- Huberize(Norm(),lower=-1,upper=2)
plot(H)
@
<<Trun, eval = TRUE, fig = TRUE>>=
T <- Truncate(Norm(),lower=-1,upper=2)
plot(T)
@
<<Min1, eval = TRUE, fig = TRUE>>=
M1 <- Maximum(Unif(0,1), Minimum(Unif(0,1), Unif(0,1)))
plot(M1)
@
<<Min2, eval = TRUE, fig = TRUE>>=
M2 <- Minimum(Exp(4),4)
plot(M2)
@
<<Min3, eval = TRUE, fig = TRUE>>=
M3 <- Minimum(Norm(2,2), Pois(3))
plot(M3)
@
%
\paragraph{Enhanced accuracy by using log scales:}
Following an idea proposed by Peter Dalgaard on {\tt r-help}, confer 
\href{https://stat.ethz.ch/pipermail/r-help/2008-September/174295.html}%
{\tt\footnotesize https://stat.ethz.ch/pipermail/r-help/2008-September/174295.html}
we use the \code{log.p} argument to be able to simulate from
the far out tails. To this end, from version 2.1, our distribution classes
gain internal slots \code{.lowerExact} and \code{.logExact} which control
whether the corresponding slot functions \code{p} and \code{q} use particular
code for computing on the log-scale respectively whether they use explicit representations
for upper quantiles (instead of computing $1-{\rm lowerQuantile}$). 
If a corresponging distribution object has slots \code{.lowerExact} and \code{.logExact}
both \code{TRUE}, we may use this to produce accurate simulations, especially for
respectively truncated distributions:
<<TruncExtr, eval = TRUE>>=
N <- Norm()
TN <- Truncate(N, 20,22)
r(TN)(20)  ## simulation accurate although :
p(N)(20, lower.tail = FALSE) ## prob that N>=20 
@

\subsection{Additional helper functions}
%
From version 1.9 on, there are methods \code{p.l} and \code{q.r} available
for \code{DiscreteDistribution} objects for the 
left-continuous variant of the cdf, i.e.; $t\mapsto {\rm p.l}(t)=P(X<t)$), and the 
right-continuous variant of the quantile function, i.e.;
$$
s\mapsto {\rm q.r}(s)=\sup\{t \,\big|\, P({\tt object}\leq t)\leq s\}
$$
From version 2.1 on, \code{p.l} and \code{q.r}, are also available for
class \code{AbscontDistribution} (where \code{q.r} takes care about 
correct treatment of gaps using helper function \code{modifyqgaps}),
for class \code{UnivarLebDecDistribution}, and for class \code{UnivarMixingDistribution}.
<<qrex, eval = TRUE>>=
B <- Binom(5,0.5)
p(B)(3)
p.l(B)(3)
q(B)(.5)
q.r(B)(0.5)
@


Again from version 2.1 on, class \code{DiscreteDistribution} has a helper
method \code{prob} which returns vector of probabilities for the 
support points. More precisely, the return value is a numeric vector 
named by the values of support points. This method is also available for
objects of class \code{UnivarLebDecDistribution} where it  returns a
two-row matrix where the column names are the values of the support points,
and the first row, named \code{"cond"}, contains the probabilities of 
the discrete part (summing up to $1$), while the second row, named \code{"abd"}
contains the probabilities of discrete part multiplied with \code{discreteWeight}; 
hence these values are the absolute probabilities of the support points.

Again for objects of class \code{UnivarLebDecDistribution}, we have methods 
\code{p.ac}, \code{d.ac}, \code{p.discrete}, \code{d.discrete}
to give the density/probability and the cumulative distribution function 
of the discrete and absolutely continuous ({\tt ac}) part of the distribution.
All these methods have an extra argument \code{CondOrAbs} with default value
\code{"cond"}, which if it does not partially match \code{"abs"}, 
returns exactly slot \code{p} (resp. \code{d}) the respective 
a.c./ discrete part of the object;  else the return value is weighted by 
the respective weight of the part, i.e. \code{acWeight}/\code{discreteWeight}.

<<probHN, eval = TRUE>>=
B0 <- as(Binom(5,0.5),"DiscreteDistribution")
   ## coercion necessary:
   ## otherwise slot "prob" of B0 will be returned
prob(B0)
HN <- Huberize(N, -2,1)
prob(HN)
@

In order to convert arbitrary univariate distributions to \code{AbscontDistribution}
from version 2.1 on, we have function \code{makeAbscontDistribution} which
takes slot \code{p} and uses \code{AbscontDistribution()} to generate a 
corresponding smoothed version; to smear out mass points on the border
of the support, these upper and lower bounds are somewhat enlarged.
Note that in the result, slots \code{p} and \code{q} are not replaced but
rather taken unchanged from the argument:

<<makeAC, eval = TRUE, fig = TRUE>>=
par(mfrow=c(2,3))
plot(makeAbscontDistribution(Nbinom(5,.5)),mfColRow=FALSE)
plot(makeAbscontDistribution(HN),mfColRow=FALSE)
par(mfrow=c(1,1))
@

Methods \code{getLow}, \code{getUp} available for classes \code{DiscreteDistribution},
\code{AbscontDistribution}, \code{UnivarLebDecDistribution}, and 
\code{UnivarMixingDistribution} return ``numerical'' end points of the respective
supports: if these distributions have finite end points, these are returned,
else a lower/upper \code{eps}-quantile is returned, where \code{ep}, by default,
is set to global options \code{TruncQuantile}.

<<getLowUp, eval = TRUE>>=
getLow(Nbinom(5,0.5))
getUp(Nbinom(5,0.5))
getLow(Norm(5,0.5))
getUp(Norm(5,0.5))
@


%
\subsection{Overloaded generic functions}
Methods \code{print}, \code{plot}, \code{show} and \code{summary} have been 
overloaded for classes \code{Distribution}, \code{Dataclass}, \code{Simulation}, 
\code{ContSimulation}, as well as \code{Evaluation} and \code{EvaluationList} to 
produce ``pretty''  output. 
More specifically there are also particular \code{show} methods
for classes \code{UnivarDistrList}, \code{UnivarMixingDistribution} and
\code{UnivarLebDecDistribution}.
%\newline 0??????\\
\code{print}, \code{plot}, \code{show} and \code{summary} have additional, 
optional arguments for plotting subsets of the simulations / results:
index vectors for the dimensions, the runs, the observations, and the 
evaluations may be passed using arguments \code{obs0},  \code{runs0}, 
\code{dims0}, \code{eval0}, confer\newline
\code{help("<mthd>-methods",package=<pkg>)} where \code{<mthd>} stands for 
\code{plot}, \code{show}, \code{print}, or \code{plot}, and \code{<pkg>} stands 
for either \pkg{distrSim} or \pkg{distrTEst}.

\subsection{Plotting}

\subsubsection[{Plotting for Distribution objects}]{Plotting for \code{Distribution} objects}
For an object of class \code{Distribution},
\code{plot} displays the density/probability function, the c.d.f.\ and the 
quantile function of a distribution. Note that all usual parameters of 
\code{plot} remain valid. For instance, you may increase the axis annotations 
and so on. \code{plot()} can also cope with \code{log}-arguments.

\paragraph{\code{xlim} argument:} More importantly, you may also 
override the automatically chosen $x$-region by passing an \code{xlim} argument:
<<cauchy1, eval = TRUE, fig = TRUE>>=
  plot(Cauchy())
@
<<cauchy2, eval = TRUE, fig = TRUE>>=
  plot(Cauchy(),xlim=c(-4,4))
@
From version 2.1, the automatic choice of the x-range of \code{plot}
for distributions has been enhanced: just as for expectation (see subsection~\ref{Expect})
we use both a quantile and a scale based approach to get sensible values.
Also   argument \code{ylim} can now be matrix-valued to use different
limits for the various panels (\code{d}, \code{p}, \code{q}, or in case of \code{UnivarLebDecDistribution},
\code{p}, \code{q}, \code{d.ac}, \code{p.ac}, \code{q.ac},
\code{d.discrete}, \code{p.discrete}, \code{q.discrete}).


\paragraph{titles:} Moreover you may control optional main, inner titles and subtitles with 
arguments \code{main} / \code{sub} / \code{inner}. To this end there are 
preset strings substituted in both expression and character vectors 
(where in the following \code{x} denotes the argument 
with which \code{plot()} was called)
\begin{itemize}
 \item[\%A] deparsed argument \code{x}
 \item[\%C] class of argument \code{x}
 \item[\%P] comma-separated list of parameter values of slot \code{param} of 
            argument \code{x}
 \item[\%Q] comma-separated list of parameter values of slot \code{param} of 
            argument \code{x} in parenthesis unless this list is empty; then 
            \code{""}
 \item[\%N] comma-separated {\tt <name> = <value>} - list of parameter values of 
            slot \code{param} of argument \code{x}
 \item[\%D] time/date at which plot is/was generated
\end{itemize}
As usual you may control title sizes and colors with 
\code{cex.main} / \code{cex.inner} / \code{cex.sub} respectively with
\code{col} / \code{col.main} / \code{col.inner} / \code{col.sub}. Additionally
it may be helpful to control top and bottom margins with arguments
\code{bmar}, \code{tmar}. 

\paragraph{step-function features:} 
We provide different default symbols for unattained [\code{pch.u}] / attained 
[\code{pch.a}] one-sided limits, which may be overridden by corresponding
arguments  \code{pch} / \code{pch.a} / \code{pch.u}.

For objects of class \code{AbscontDistribution}, you may set the number of grid 
points used by an \code{ngrid} argument; also the ``quantile''-panel
takes care of finite left/right endpoints of support and optionally tries
to identify constancy region of the \code{p}-slot. 

For objects of class \code{DiscreteDistributions}, we use \code{stepfun()} from
package \pkg{base} as far as possible and (also for panel ``q'' for
\code{AbscontDistributions}) consequently take over its arguments 
\code{do.points}, \code{verticals}, \code{col.points} / \code{col.vert} / 
\code{col.hor} and \code{cex.points}.

As examples consider the following plots: The standard plot for
a discrete distribution is shown in Figure~\ref{plotexPic1}.

\begin{figure}[p] 
\begin{center}
<<plotex1, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
plot(Binom(size = 4, prob = 0.3))
@
    \caption{\label{plotexPic1}{Standard plot for discrete distributions
    }}
\end{center}
\end{figure}

Omitting the point symbols at jump points (\code{do.points = FALSE})
and the corresponding vertical lines (\code{verticals = FALSE}) 
in the \code{p} and \code{q} panels
gives  Figure~\ref{plotexPic2}.
\begin{figure}[p] 
\begin{center}
<<plotex2, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
plot(Binom(size = 4, prob = 0.3), do.points = FALSE, verticals = FALSE)
@
    \caption{\label{plotexPic2}{Plot for discrete distributions without
    extra symbols at jump points and vertical lines
    }}
\end{center}
\end{figure}

Instead we might use a somewhat enlarged (\code{cex.main = 1.6}) 
main title (\code{main = TRUE}) and omit the 
panel titles (\code{inner = FALSE}). To this end, we should increase the
margin between main title and panels (by \code{tmar = 6}). This
is shown in Figure~\ref{plotexPic3}.

\begin{figure}[p] 
\begin{center}
<<plotex3, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
plot(Binom(size = 4, prob = 0.3), main = TRUE, inner = FALSE, cex.main = 1.6,
     tmar = 6)
@
    \caption{\label{plotexPic3}{Plot for discrete distributions using a main 
    title
    }}
\end{center}
\end{figure}

Changing point sizes (\code{cex.points}) and line width (\code{lwd}),
and using (default) panel titles (set \code{TRUE} by default) 
gives a somewhat different picture
as in Figure~\ref{plotexPic4}.
     
\begin{figure}[p]
\begin{center}
<<plotex4, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
plot(Binom(size = 4, prob = 0.3), cex.points = 1.2, pch = 20, lwd = 2)
@
    \caption{\label{plotexPic4}{Plot for discrete distributions using panel 
    titles and changed point sizes
    }}
\end{center}
\end{figure}

Different colors for different plot elements can be used by
arguments \code{col} (general), \code{col.points} (jump points),
\code{col.sub} (sub-titles), \code{col.inner} (panel titles).
In Figure~\ref{plotexPic5}, this is shown; the figure uses main
(\code{main = TRUE}),
inner (\code{TRUE} by default) and sub (\code{sub = TRUE}) titles,
using the default titles respectively.
     
\begin{figure}[p]
\begin{center}
<<plotex5, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
B <- Binom(size = 4, prob = 0.3)
plot(B, col="red", col.points = "green", main = TRUE, col.main="blue",
     col.sub = "orange", sub = TRUE, cex.sub = 0.6, col.inner = "brown")
@
    \caption{\label{plotexPic5}{Plot for discrete distributions using main, panel 
    and sub titles with changed colors
    }}
\end{center}
\end{figure}

Changing plot size for marking the jump points by \code{cex.points = 1.2}
and selecting symbols for left limits (by \code{pch.u = 20}) and
right values (by \code{pch.a = 20}) at jump points is exemplified  in 
Figure~\ref{plotexPic6}.

\begin{figure}[p]
\begin{center}
<<plotex6, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
plot(Nbinom(size = 4,prob = 0.3), cex.points = 1.2, pch.u = 20, pch.a = 10)
@
    \caption{\label{plotexPic6}{Plot for discrete distributions with different
    symbols for marking jump points
    }}
\end{center}
\end{figure}

Using log-scale for both axes by \code{log = "xy"} and a reduced number of grid
points for plotting by \code{ngrid = 200} is shown in 
Figure~\ref{plotexPic7}.

\begin{figure}[p]
\begin{center}
<<plotex7, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
plot(Chisq(), log = "xy", ngrid = 100)
@
    \caption{\label{plotexPic7}{Plot for absolutely continuous distributions 
    using log scales
    }}
\end{center}
\end{figure}

Changing line type by \code{lty = 3} and color by \code{col = "red"},
axis notation orientation by \code{las = 2} and the number of grid
points used for plotting by \code{ngrid = 200} is shown in 
Figure~\ref{plotexPic8}.
\begin{figure}[p] 
\begin{center}
<<plotex8, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
plot(Norm(), lwd=3, col = "red", ngrid = 200, lty = 3, las = 2)
@
    \caption{\label{plotexPic8}{Plot for absolutely continuous distributions 
    with different line type and reduced number of grid points
    }}
\end{center}
\end{figure}

You may also use hook functions in distribution plots from version 2.0 on,
which is especially useful for plotting grids in the background as shown in  
Figure~\ref{plotexPic9}; this plot also has non-default titles using
preset strings substitutions (\%N, \%C, \%P, \%A, \%D).

\begin{figure}[p]
\begin{center}
<<plotex9, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
plot(Norm(), panel.first = grid(), main = "my Distribution: %A",
     inner = list(expression(paste(lambda, "-density of %C(%P)")), "CDF",
                  "Pseudo-inverse with param's %N"),
     sub = "this plot was correctly generated on %D",
     cex.inner = 0.9, cex.sub = 0.8)
@
    \caption{\label{plotexPic9}{Plot for absolutely continuous distributions 
    using non-standard titles and with a background grid
    }}
\end{center}
\end{figure}

Special care is take to correctly depict jumps in the quantile
function / gaps in the support; from version 2.0 on you have
function \code{setgaps()} to automatically such gaps, but in case
of very small density values this may give some ``false positives''
as shown for the $\chi^2$ distribution in Figure~\ref{plotexPic10}

\begin{figure}[p] 
\begin{center}
<<plotex10, eval = TRUE, fig = TRUE, width=8.0,height=4.5>>=
Ch <- Chisq(); setgaps(Ch, exactq = 3)
plot(Ch, cex = 1.2, pch.u = 20, pch.a = 10, col.points = "green", 
     col.vert = "red")
@
    \caption{\label{plotexPic10}{Plot for absolutely continuous distributions 
    with automatic gap detection
    }}
\end{center}
\end{figure}

From version 2.0 on, you may override the given panel
configuration using argument \code{mfColRow=FALSE} 
---see Figure~\ref{plotexPic11}.

\begin{figure}[p]
\begin{center}
<<plotex11, eval = TRUE, fig = TRUE, width=8.0,height=8.0>>=
layout(matrix(c(1,3,2,3), nrow=2))
plot(N, mfColRow = FALSE)
@
    \caption{\label{plotexPic11}{Plot for absolutely continuous distributions 
    using non-standard panel configuration
    }}
\end{center}
\end{figure}

Following a suggestion by \href{mailto:unwin@math.uni-augsburg.de}{Anthony Unwin},
from version 2.1 you may also select the panels you like to plot, using
argument \code{to.draw.arg}; the corresponding panels are named and
may either be given by name or by number (the rank in drawing the default
``complete'' plot); for details
see \code{?plot}.  As example for this panel selection, see 
Figure~\ref{plotexPic12}.

\begin{figure}[p] 
\begin{center}
<<plotex12, eval = TRUE, fig = TRUE, width=8.0,height=8.0>>=
layout(matrix(c(rep(1,6),2,2,3,3,4,4,5,5,5,6,6,6), 
                   nrow=3, byrow=TRUE))
plot(HN, mfColRow = FALSE,
        to.draw.arg=c("p","d.c","p.c","q.c", "p.d","q.d"))
@
    \caption{\label{plotexPic12}{Plot for Lebesgue decomposed distributions 
    with user-chosen selection of plotted panels
    }}
\end{center}
\end{figure}

\subsubsection[{Plotting for Dataclass objects}]{Plotting for \code{Dataclass} objects}
For objects of  class \code{Dataclass} ---or of a corresponding subclass---
 \code{plot} plots the sample against the run index and in case  of  
 \code{ContSimulation} the contaminating variables are highlighted by a 
 different color. Additional arguments controlling
 the plot as in the default \code{plot} command may be passed, 
 confer \code{help("plot-methods",package="distrSim")}.
\subsubsection[{Plotting for Evaluation objects}]{Plotting for \code{Evaluation} objects}
 For an object of class \code{Evaluation},
\code{plot} yields a boxplot of the results of the evaluation.
For an object of class \code{EvaluationList},
\code{plot} regroups the list according to the different columns/coordinates of 
the result of the evaluation; for each such coordinate, a boxplot is generated, 
containing possibly several procedures, and, if evaluated at a 
\code{Contsimulation}, the plots are also grouped into evaluations on ideal and
real data. As for the usual \code{boxplot} function you may pass additional 
``\code{plot}-type'' arguments to this particular \code{plot} method, confer 
\code{help("plot-methods",package="distrTEst")}. In particular, the 
\code{plot}-arguments \code{main} and \code{ylim}, however, may also be 
transmitted coordinatewise, i.e.; a vector of the same length as the dimension 
of the result {\tt resDim} (e.g.\ parameter dimension), respectively a 
{\tt 2 x resDim} matrix, or they may be transmitted globally, using the 
usual {\tt S} recycling rules.
%\newline??????1

\subsubsection[{Plotting for L2paramFamily objects}]{Plotting for \code{L2paramFamily} objects}
In package \pkg{distrMod} we have an additional plotting method for class
\code{L2paramFamily}; besides the underlying model distribution this also
plots the coordinates of the $L_2$-derivative (scores function).
From version 2.1 on, this plot is as flexible as the one for \code{Distribution}.
You can select the panels to be plotted by argument \code{to.draw.arg},
and may use (almost) all arguments generally available for \code{plot}.
In particular, also you may use an argument \code{panel.first=grid()} to 
produce a grid behind the panel. 


\subsection[liesInSupport]{\code{liesInSupport}}
For all discrete distribution classes, we have methods \code{liesInSupport} to 
check whether a given vector/ a matrix of points lies in the support of the 
distribution.

\subsection[Simulation (in package distrSim)]%
{Simulation (in package \pkg{distrSim})}
%
From version 1.6 on, \code{simulation} is available in package  \pkg{distrSim}.

For the classes \code{Simulation} and \code{ContSimulation}, we normally will
not save the current values of the simulation, as they can easily be reproduced
knowing the values of the other slots of this class.
%
So when declaring a new object of either of the two classes, the slot 
\code{Data} will be empty (\code{NULL}).
To fill it with the simulated values, we have to apply the method 
\code{simulate} to the object. This has to be redone whenever another slot of 
the object is changed.
%
To guarantee reproducibility, we use the slot \code{seed}.\\
%
This slot is controlled and set through 
\href{mailto:pgilbert@bank-banque-canada.ca}{Paul Gilbert's} \pkg{setRNG} 
package.
By default, \code{seed} is set to \code{setRNG()}, which returns the current 
``state'' of the random number generator (RNG). So the user does not need to 
specify a value for \code{seed}, and nevertheless may reproduce his samples: 
He simply uses \code{simulate} to fill the \code{Data} slot.
If the user wants to, he may also set the \code{seed} explicitly via the 
replacement function \code{seed()}, but has to take care of the correct format 
himself, confer the documentation of \code{setRNG}. One easy way to fill 
the \code{Data} slot of a simulation \code{X} with ``new'' random numbers is
<<simulate, eval = TRUE>>=
X <- Simulation()
seed(X) <- setRNG()
simulate(X)
Data(X)[1:10]
@
%
\subsection[Evaluate (in package distrTEst)]%
{Evaluate (in package \pkg{distrTEst})}\label{evaluate}
%
From version 1.6 on \code{evaluate} is available in  \pkg{distrTEst}.

In an object of class \code{Evaluation}  we store relevant information
about an evaluation of a statistical procedure (estimator/test/predictor)
on an object of class \code{Dataclass}, including the concrete results of
this evaluation. An object of class \code{Evaluation}  is generated by an 
application of method \code{evaluate} which takes as arguments an object of 
class  \code{Dataclass} and a procedure of type \code{function}. As an example, 
confer Example~\ref{simex}.
For data of class \code{Contsimulation}, the result takes a slightly different,
combining evalations on ideal and real data.
%
\subsection{Is-Relations}
%
By means of \code{setIs}, we have ``told'' {\sf R}  that a distribution object 
\code{obj} of class
\begin{itemize}
  \item  \code{"Unif"} with  $\code{Min} \doteq 0$ and  $\code{Max} \doteq 1$ 
   also is a Beta distribution with parameters \code{shape1 = 1, shape2 = 1}
  \item  \code{"Geom"} also is a negative Binomial distribution with parameters
         \code{size = 1, prob = prob(obj)}
  \item \code{"Cauchy"} with $\code{location} \doteq 0$ and 
         $\code{scale}\doteq 1$ also is a T distribution with parameters
         \code{df = 1, ncp = 0}
  \item \code{"Exp"} also is a Gamma distribution with parameters 
         \code{shape = 1, scale = 1/rate(obj)} and
         a Weibull  distribution with parameters 
         \code{shape = 1, scale = 1/rate(obj)}
  \item \code{"Chisq"} with non-centrality $\code{ncp}\doteq 0$ also is a 
        Gamma distribution with parameters \code{shape = df(obj)/2, scale = 2}
  \item \code{"DiscreteDistribution"}  (from version 1.9 on) with an equally 
         spaced support also is a \code{"LatticeDistribution"}
\end{itemize}
%
\subsection{Further methods}
%
When iterating/chaining mathematical operations on a univariate distribution,
generation process of random variables can become clumsy and slow.
To cope with this, we introduce a sort of ``Forget-my-past''-method 
\code{simplifyr} that replaces the  chain of mathematical operations in 
the \code{r}-method by drawing with replacement from a large 
sample ($10^{\tt RtoDPQ.e}$) of these.
%
\subsection[Functionals (in package distrEx)]%
{Functionals (in package \pkg{distrEx})}\label{Functionals}
%
\subsubsection{Expectation}\label{Expect}
The most important contribution of package \pkg{distrEx} is a general 
expectation operator. In basic statistic courses, the expectation 
${\rm E}$ may come as ${\rm E}\,[X]$, ${\rm E}\,[f(X)]$, ${\rm E}\,[X|Y=y]$,
or ${\rm E}\,[f(X)|Y=y]$. Our operator (or in S4-language ``generic function'') 
\code{E} covers all of these situtations (or {\it signatures\/}).
\paragraph{default call}
The most frequent call will be \code{E(X)} where \code{X} is an (almost) 
arbitrary distribution object.
More precisely, if \code{X} is of a specific distribution class like 
\code{Pois}, it is evaluated exactly using analytic terms. Else if it is of 
class \code{DiscreteDistribution} we use a sum over the support of \code{X},
and if it is of class \code{AbscontDistribution} we use numerical 
integration\footnote{i.e., we first try (really(!): \code{try})
\code{integrate} and if this fails we use Gau{\ss}-Legendre integration 
according to \cite{NumR:92}, see also \code{?distrExIntegrate}}; 
for {\tt X} of class \code{UnivarLebDecDistribution}, expectations
for discrete and absolutely continuous part are evaluated separately and
subsequently combined according to their respective weights.
If we only  know that {\tt X} is of class \code{UnivariateDistribution} we 
use Monte-Carlo 
integration. This also is the default method in for class 
\code{MultivariateDistribution}, while for \code{DiscreteMVDistribution} we 
again use sums. For an object \code{Y} of a subclass of class 
union \code{AffLinDistribution}, we determine the expectation as 
\code{Y@a * E(Y@X0) + Y@b} and hence use analytic terms for \code{X0} if
available.

\paragraph{with a function as argument}
 we proceed just as without: if \code{X} is of class\linebreak[4]
 \code{DiscreteDistribution}, we use a sum over the support of \code{X},
and if \code{X} is of class\linebreak[4] \code{AbscontDistribution} we use numerical 
integration; else we use Monte-Carlo integration.
\paragraph{in addition: with a condition as argument} we simply use the 
corresponding \code{d} respective \code{r} slots with the additional 
argument \code{cond}.
\paragraph{exact evaluation}
is available for \code{X} of class \code{Arcsine}, \code{Beta} (for noncentrality $0$), 
\code{Binom}, \code{Cauchy}, \code{Chisq}, \code{Dirac}, \code{Exp}, \code{Fd}, 
\code{Gammad}, \code{Geom}, \code{Gumbel}, \code{Hyper}, \code{Logis}, \code{Lnorm}, 
\code{Nbinom}, \code{Norm}, \code{Pareto}, \code{Pois}, \code{Td}, \code{Unif}, \code{Weibull}.
\paragraph{examples} $ \mbox{ }$\newline
%<<expectationPrep, eval = TRUE, echo=FALSE>>=
%SweaveListingOptions("inSweave"=TRUE)
%@
<<expectation, eval = TRUE>>=
D4 <- LMCondDistribution(theta = 1)
D4  # corresponds to Norm(cond, 1)
N <- Norm(mean = 2)

E(D4, cond = 1)
E(D4, cond = 1, useApply = FALSE)
E(as(D4, "UnivariateCondDistribution"), cond = 1)
E(D4, function(x){x^2}, cond = 2)
E(D4, function(x){x^2}, cond = 2, useApply = FALSE)
E(N, function(x){x^2})
E(as(N, "UnivariateDistribution"), function(x){x^2}, 
     useApply = FALSE) # crude Monte-Carlo
E(D4, function(x, cond){cond*x^2}, cond = 2,
  withCond = TRUE)
E(D4, function(x, cond){cond*x^2}, cond = 2,
  withCond = TRUE, useApply = FALSE)
E(N, function(x){2*x^2})
E(as(N, "UnivariateDistribution"), function(x){2*x^2},
  useApply = FALSE) # crude Monte-Carlo
Y <- 5 * Binom(4, .25) - 3
Y
E(Y)  
@
%
\paragraph{Controlling integration range:} From version 2.1 on, \code{E} gains
arguments \code{low} and \code{upp} to restrict evaluation of the
integrand to a given integration domain; these arguments can also
be passed through by other functionals based on expectation, like
\code{var}, \code{sd}, \code{skewness}, and \code{kurtosis}.
<<expectationlow, eval = TRUE>>=
E(Cauchy(), low=3, upp=5)
var(Cauchy(), low=3, upp=5)
@

\paragraph{Controlling accuracy:} From version 2.1 on, our 
expectation methods gain explicit arguments to set accuracy locally;
i.e.; the MC-methods have an argument \code{Nsim} defaulting to 
global option \code{MCIterations}, while the methods for 
class \code{AbscontDistribution} using numerical
integration have an argument \code{rel.tol} defaulting to global option
\code{ErelativeTolerance}. To obtain a sensible integration range
automatically, these methods use both quantile and scale based methods;
more precisely you may pass on arguments \code{lowerTruncQuantile}
and \code{upperTruncQuantile}, defaulting to global options \code{ElowerTruncQuantile}, 
and \code{EupperTruncQuantile}, respectively, by means of these we determine
lower and upper quantiles $l_0$, $u_0$. In addition, we determine scale based 
bounds as ${\rm median} \pm s_f \,{\rm IQR}$ where $s_f$ is a scaling factor
to be passed on as argument \code{IQR.fac} which defaults to global option
\code{IQR.fac}.
%<<expectationPrep, eval = TRUE, echo=FALSE>>=
%SweaveListingOptions("inSweave"=TRUE)
%@
<<expectation2, eval = TRUE>>=
E(N, function(x)x^2) 
E(N, function(x)x^2,  lowerTruncQuantile = 1e-5)
var(Cauchy(), low =3, upperTruncQuantile = 1e-5,  IQR.fac = 10)
var(Cauchy(), low =3, upperTruncQuantile = 1e-10, IQR.fac = 20)
@

%
\subsubsection{Variance}
The next-common functional is the variance. In order to keep a unified
notation we will use the same name as for the empirical variance, i.e.,
 \code{var}.
\paragraph{masking \pkg{stats}-method \code{var}}
To cope with the different argument structure of the empirical variance,
i.e. 
\code{var(x, y = NULL, na.rm = FALSE, use)}  and our functional variance, 
i.e., 
\code$var(x, fun = function(t) {t}, cond, withCond = FALSE, useApply = TRUE, ...)$
we have to mask the original \pkg{stats}-method:
<<var, eval = FALSE>>=
  var <- function(x , ...)
       {dots <- list(...)
        if(hasArg(y)) y <- dots$"y"
        na.rm <- ifelse(hasArg(na.rm), dots$"na.rm", FALSE)
        if(!hasArg(use))
             use <- ifelse (na.rm, "complete.obs","all.obs")
        else use <- dots$"use"
        if(hasArg(y))
           stats::var(x = x, y = y, na.rm = na.rm, use)
        else
           stats::var(x = x, y = NULL, na.rm = na.rm, use)
        }
@
before registering \code{var} as generic function.
Doing so, if the \code{x} (or the first) argument of \code{var}
is not of class \code{UnivariateDistribution}, \code{var} behaves identically to
the \pkg{stats} package
\paragraph{default method} if \code{x} is of class \code{UnivariateDistribution}, 
\code{var} just returns the variance of distribution \code{X} --- or 
of \code{fun(X)} if a function is passed as argument fun, or, if a condition 
argument \code{cond} (for $Y=y$),  ${\rm Var}\,[X|Y=y]$ respectively 
${\rm Var}\,[f(X)|Y=y]$ --- just as for \code{E}.
The same goes for corresponding arguments controlling the accuracy 
of \code{E} locally from version 2.1 on (see paragraph ``Controlling accuracy''):
These may simply passed through in a call to \code{var}.  

For an object \code{Y} of a subclass of class 
union \code{AffLinDistribution}, we determine the variance as 
\code{Y@a\textasciicircum2 * var(Y@X0)} and hence use analytic terms for \code{X0} if
available.
\paragraph{exact evaluation} is provided for specific distributions if no 
function and no condition argument is given:
this is available for \code{X} of class \code{Arcsine}, \code{Beta} (for noncentrality $0$), 
\code{Binom}, \code{Cauchy}, \code{Chisq}, \code{Dirac}, \code{Exp}, \code{Fd}, 
\code{Gammad}, \code{Geom}, \code{Gumbel}, \code{Hyper}, \code{Logis}, \code{Lnorm}, 
\code{Nbinom}, \code{Norm}, \code{Pareto}, \code{Pois}, \code{Unif}, \code{Td}, \code{Weibull}.
\subsubsection{Further functionals}
By the same techniques we provide the following functionals for univariate 
distributions:
\begin{itemize}
  \item standard deviation: \code{sd}
  \item skewness: \code{skewness}  
  (code contributed by G. Jay Kerns, {\small \tt gkerns@ysu.edu})
  \item kurtosis: \code{kurtosis} 
   (code contributed by G. Jay Kerns, {\small \tt gkerns@ysu.edu})
  \item median: \code{median} (not for function arguments)
  \item median of absolute deviations: \code{mad} (not for function/condition 
  arguments)
  \item interquartile range: \code{IQR} (not for function arguments)
\end{itemize}
For details, see \code{?skewness}.
%
\subsection[Truncated moments (in package distrEx)]{Truncated moments 
(in package \pkg{distrEx})}\label{m1df}
%
For Robust Statistics, the first two truncated moments are very useful. These 
are realized as generic functions \code{m1df} and \code{m2df}: They use the 
expectation operator for general univariate distributions, but are overloaded 
for most specific distributions:
\begin{itemize}
  \item \code{Binom}
  \item \code{Pois}
  \item \code{Norm}
  \item \code{Exp}
  \item \code{Chisq}
\end{itemize}
%
\subsection[Distances (in package distrEx)]{Distances (in package %
 \pkg{distrEx})}\label{Distances}
%
For several purposes like Goodness-of-fit tests or minimum-distance estimators, 
distances between distributions are useful. This applies in particular to Robust 
Statistics. In package \pkg{distrEx}, we provide the follwoing distances:
\begin{itemize}
  \item Kolmogoroff distance (\code{KolmogorovDist})
  \item total variation distance (\code{TotalVarDist})
  \item Hellinger distance   (\code{HellingerDist})
  \item Cram\'er von Mises distance  (\code{CvMDist}) with an additional
  argument for the weighting measure $\mu$ (defaulting to second operand $Q$):
  $$
  d_\mu(P,Q)^2=\int \Big(Q((-\infty;t])-P((-\infty;t])\Big)^2\,\mu(dt)
  $$
  \item convex-contamination ``distance'' (asymmetric!) (\code{ContaminationSize}) defined as
  $$
d(Q,P):=\inf\{r>0\,|\, \exists \;\mbox{probability } H:\quad Q=(1-r)P+rH \}
  $$
  \item from version 2.1 on: an asymmetric version of total variation distance
  (\code{AsymTotalVarDist}):
  to given ratio $\rho\ge 0$ of negative to positive part of the deviation we 
  set 
  $$
d_{v;\rho}(Q,P):=\int (dQ - c\,dP)_+ 
  $$
where $c\in\R $ is such that
$$\rho \int (dQ - c\,dP)_+=\int (dQ - c\,dP)_-$$
  \item minimal total variation distance   (\code{OAsymTotalVarDist}):
  $$
d_{v;{\rm opt}}(Q,P):= \min_c \int (dQ - c\,dP)_+ +  (dQ - c\,dP)_-
  $$
\end{itemize}
Methods using numerical integration use a similar technique as
for the expectation mentioned in subsection~\ref{Expect}, combining
scale and quantile based methods to obtain a sensible integration
range automatically.
 
\subsection[Functions for demos (in package distrEx)]{Functions for demos 
(in package \pkg{distrEx})}\label{Demos}
%
To illustrate the possibilities with packages \pkg{distr} and \pkg{distrEx} we 
include two major demos to \pkg{distrEx}, each with extra code to it ---
one for the CLT and one for the LLN.

From version 2.0 on, we have started a new package \pkg{distrTeach}, which
is to use the capabilities of packages \pkg{distr} and \pkg{distrEx} for
illustrating topics of Stochastics and Statistics as taught in secondary
school. So far we have moved the illustrations for the CLT and the LLN 
just mentioned to it.

\subsubsection{CLT for arbitrary summand distribution}

By means of our convolution algortithm as well as with the operators \code{E} 
and \code{sd} an illustration for the CLT is readily written: 
function \code{illustrateCLT}, respectively demo \code{illustCLT}.
For plotting, we have particular methods for discrete and absolute continuous 
distributions. The user may specify a given summand distribution, an upper 
limit for the consecutive sums to be considered and a pause between the 
corresponding plots in seconds. From version 1.9 on, we also include a 
\code{TclTk}-based version of this demo, where the user may enter the 
distribution argument (i.e.; the summands' distribution) into a text line and 
control the sample size by a slider in some widget: \code{illustCLT\_tcl}
From version 2.0 on, this functionality has moved to package \pkg{distrTeach}.

\subsubsection{LLN for arbitrary summand distribution}

From version 1.9 on, similarly, we provide an illustration for the LLN:
function \code{illustrateLLN}, respectively demo \code{illustLLN}.
The user may specify a vector of sample sizes to be 
considered, the number of replicates to be drawn and a pause between the 
corresponding plots in seconds, also, optionally, the limiting expectation
(in case of class \code{Cauchy}: the non-limiting median) is drawn as a line and
Chebyshev/CLT-based (pointwise) confidence bands and their respective empirical 
coverages are displayed. 
From version 2.0 on, this functionality has moved to package \pkg{distrTeach}.

\subsubsection{Deconvolution example}

To illustrate conditional distributions and their implementation in 
\linebreak[4]\pkg{distrEx}, we consider the following situation:
We consider a signal $X\sim P^X$ which is disturbed by noise 
$\varepsilon\sim P^\varepsilon$, independent from $X$; in fact we observe
$Y=X+\varepsilon$ and want to reconstruct $X$ by means of $Y$. By means of the 
generating function \code{PrognCondDistribution}
of package \pkg{distrEx}, for absolutely continuous $P^X, P^\varepsilon$, we may 
determine the factorized conditional distribution $P^{X|Y=y}$, and based on this 
either its (posterior) mode oder (posterior) expectation; also see
\code{demo(Prognose, package="distrEx")}.

% -----------------------------------------------------------------------------
\section[Package distrMod]{Package {\tt distrMod}}\label{distrMod}
% -----------------------------------------------------------------------------
The package \pkg{distrMod} aims for an object orientated (S4-styple) 
implementation of probability models and introduces several new S4-classes
for this purpose. Moreover, it includes functions to compute minimum criterion 
estimators -- in particular, minimum distance and maximum likelihood (i.e., 
minimum negative log-likelihood) estimators.
% -----------------------------------------------------------------------------
\subsection{Symmetry Classes}
% -----------------------------------------------------------------------------
As symmetry is a property which usually cannot be proven via numerical 
computations, we introduce the S4-class \code{Symmetry} and corresponding
subclasses which may serve as slots which indicate that there exists a
certain symmetry. So far, we have subclasses for the symmetry of distributions
as well as for the symmetry of functions; confer Figure~\ref{figSym}.
\ifpdf
\begin{figure}[!ht]
\vspace{2ex}
  \begin{center}
    \includegraphics[width=13cm]{Symmetry.pdf}%
    \caption{\label{figSym}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{Symmetry} where we do not
    repeat inherited slots
    }}
  \end{center}
\vspace{-4ex}
\end{figure}
\else
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=7.5cm]{Symmetry.eps}%
    \caption{\label{figSym}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{Symmetry} where we do not 
    repeat inherited slots
    }}
  \end{center}
\vspace{-1ex}
\end{figure}
\fi
% -----------------------------------------------------------------------------
\newpage
\subsection{Model Classes}
% -----------------------------------------------------------------------------
Based on class \code{Distribution} and its subclasses we define classes 
for families of probability measures. So far, we specialised this to 
parametric families of probability measures; confer Figure~\ref{figPF}. But it 
would also be possible to derive subclasses for other (e.g., semiparametric) 
families of probability measures. In case of $L_2$-differentiable (i.e., 
smoothly parameterized) parametric families we introduce several additional
slots, in particular the slot \code{L2deriv} which is of class \code{EuclRandVarList}.
Hence, package \pkg{distrMod} depends on package \pkg{RandVar}~\cite{MK:05}.
\ifpdf
\begin{figure}[!ht]
\vspace{2ex}
  \begin{center}
    \includegraphics[width=10cm]{ProbFamily.pdf}%
    \caption{\label{figPF}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{ProbFamily} where we do not
    repeat inherited slots
    }}
  \end{center}
\vspace{-4ex}
\end{figure}
\else
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=7.5cm]{ProbFamily.eps}%
    \caption{\label{figPF}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{ProbFamily} where we do not 
    repeat inherited slots
    }}
  \end{center}
\vspace{-1ex}
\end{figure}
\fi
%
Note that for general central distribution in the one-dimensional 
location and scale model, we need to determine a consistency factor
for the {\tt MAD} (when used as scale estimator). From version 2.1 on
this consistency factor is determined automatically.
% -----------------------------------------------------------------------------
\subsection[Parameter in a parametric family]%
{Parameter in a parametric family: class \code{ParamFamParameter}}
% -----------------------------------------------------------------------------
In many applications, it is not the whole parameter of a parametric family which
is of interest, but rather parts of it, while the rest of it either is known
and fixed or has to be estimated as a nuisance parameter; in other situations,
we are interested in a (smooth) transformation of the parameter. This all
is realized in a class design for the parameter of a parametric family 
---class \code{ParamFamParameter}, the formal class of a slot of class
\code{ParamFamily}.
\ifpdf
\begin{figure}[!ht]
\vspace{2ex}
  \begin{center}
    \includegraphics[width=4cm]{ParamFamParameter.pdf}%
    \caption{\label{figPFParam}{\footnotesize Inheritance relations and slots of 
    \code{ParamFamParameter} where we do not repeat inherited slots
    }}
  \end{center}
\vspace{-4ex}
\end{figure}
\else
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=7.5cm]{ParamFamParameter.eps}%
    \caption{\label{figPFParam}{\footnotesize Inheritance relations and slots of 
    \code{ParamFamParameter} where we do not repeat inherited slots
    }}
  \end{center}
\vspace{-1ex}
\end{figure}
\fi
It has slots \code{name} (the name of the parameter), \code{main} (the interesting
aspect of the parameter), \code{nuisance} an unknown part of the parameter of
secondary interest, but which has to be estimated, for instance for confidence
intervals, and \code{fixed} a known and fixed part of the parameter. Besides
these it also has a slot \code{trafo} which also sort of
  arises in class \code{Estimate}.

  \code{trafo} realizes partial influence curves; i.e.; we are only
  interested is some possibly lower dimensional smooth (not necessarily
  linear or even coordinate-wise) aspect/transformation $\tau$
  of the parameter $\theta$.

  To be coherent with the corresponding {\em nuisance}
  implementation, we make the following convention:

  The full parameter $\theta$ is split up coordinate-wise
  in a main parameter $\theta'$ and a nuisance parameter
  $\theta''$ (which is unknown, too, hence has to be
  estimated, but only is of secondary interest) and a fixed,
  known part $\theta'''$.
  
  Without loss of generality, we restrict ourselves to the case that
  transformation $\tau$ only acts on the main parameter
  $\theta'$ --- if we want to transform the whole
  parameter, we only have to assume both nuisance parameter
  $\theta''$ and fixed known part of the parameter $\theta'''$
  have length 0.
  
  \paragraph{To the implementation:} 
  
  Slot \code{trafo} can either contain a (constant) matrix
  $D_\theta$ or a function
  $$\tau\colon \Theta' \to \tilde \Theta,\qquad \theta \mapsto \tau(\theta)$$%
mapping main parameter
  $\theta'$ to some range $\tilde \Theta$.

  If {\em slot value} \code{trafo} is a function, besides $\tau(\theta)$, 
  it will also return the corresponding derivative matrix
  $\frac{\partial}{\partial \theta}\tau(\theta)$. 
  More specifically, the return value of this function \code{theta} is a
  list with entries \code{fval}, the function value $\tau(\theta)$,
  and \code{mat}, the derivative matrix.

  In case \code{trafo} is a matrix $D$, we interpret it as such a derivative
  matrix $\frac{\partial}{\partial \theta}\tau(\theta)$,
  and, correspondingly, $\tau(\theta)$ is the linear mapping
  $\tau(\theta)=D\,\theta$.\par

  According to the signature, {\em method} \code{trafo} will return different
  return value types. For signatures
  \code{Estimate,missing}, \code{Estimate,ParamFamParameter}, and
    \code{ParamFamily,ParamFamParameter}, it will return a list with entries
    \code{fct}, the function  $\tau$, and \code{mat}, the matrix
    $\frac{\partial}{\partial \theta}\tau(\theta)$.
    function $\tau$ will then return the list \code{list(fval, mat)}
    mentioned above.  For signatures \code{ParamFamily,missing} and 
    \code{ParamFamParameter,missing}, it will just return the
     corresponding matrix.

    From version 2.1 on, there are helper functions
\code{trafo.fct()} (see \code{?trafo.fct}) and \code{trafoEst}.
While \code{trafo.fct()}  allows to access  ``function'' aspect of the 
transformation, returning the corresponding function, \code{trafoEst} 
transforms an existing estimator of class \code{estimate}
  consistently (i.e.; with corresponding \code{untransfromed.estimate}
   and \code{untransformed.asvar} information and transformed \code{asvar}) 
   by a ``trafo'' function; see \code{?trafoEst}.

% -----------------------------------------------------------------------------
%\newpage
\subsection{Risk Classes}
% -----------------------------------------------------------------------------
The risk classes are up to now (i.e, version 2.0) not used inside of the 
distr-family. They are however used in the RobASt-family~\cite{MK:05}. We
distinguish between various finite-sample and asymptotic risks; confer 
Figure~\ref{figRT}. The bias and norm classes given in Figure~\ref{figBT} 
and Figure~\ref{figNT}, respectively, occur as slots of the risk classes.
\ifpdf
\begin{figure}[!ht]
\vspace{2ex}
  \begin{center}
    \includegraphics[width=7cm]{BiasType.pdf}%
    \caption{\label{figBT}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{BiasType} where we do not
    repeat inherited slots
    }}
  \end{center}
\vspace{-4ex}
\end{figure}
\else
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=6cm]{BiasType.eps}%
    \caption{\label{figBT}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{BiasType} where we do not 
    repeat inherited slots
    }}
  \end{center}
\vspace{-1ex}
\end{figure}
\fi
\ifpdf
\begin{figure}[!ht]
\vspace{2ex}
  \begin{center}
    \includegraphics[width=4cm]{NormType.pdf}%
    \caption{\label{figNT}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{NormType} where we do not
    repeat inherited slots
    }}
  \end{center}
\vspace{-4ex}
\end{figure}
\else
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=7.5cm]{NormType.eps}%
    \caption{\label{figNT}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{NormType} where we do not 
    repeat inherited slots
    }}
  \end{center}
\vspace{-1ex}
\end{figure}
\fi
\ifpdf
\begin{figure}[!ht]
\vspace{2ex}
  \begin{center}
    \includegraphics[width=13cm]{RiskType.pdf}%
    \caption{\label{figRT}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{RiskType} where we do not
    repeat inherited slots
    }}
  \end{center}
\vspace{-4ex}
\end{figure}
\else
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=7.5cm]{RiskType.eps}%
    \caption{\label{figRT}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{RiskType} where we do not 
    repeat inherited slots
    }}
  \end{center}
\vspace{-1ex}
\end{figure}
\fi
% -----------------------------------------------------------------------------
\clearpage
\subsection{Minimum Criterion Estimation}
% -----------------------------------------------------------------------------
The S4-classes and methods defined inside of our distr-family enable us to
define general functions for the computation of minimum criterion estimators
-- in particular, minimum distance and maximum likelihood (i.e., minimum 
negative log-likelihood) estimators. The main function for this purpose is
\code{MCEstimator}. As an example we can use the negative log-likelihood as
criterion; i.e., compute the maximum likelihood estimator.
<<MCEstimator, eval = TRUE>>=
    library(distrMod)
    x <- rgamma(50, scale = 0.5, shape = 3)
    G <- GammaFamily(scale = 1, shape = 2)
    negLoglikelihood <- function(x, Distribution){
        res <- -sum(log(Distribution@d(x)))
        names(res) <- "Negative Log-Likelihood"
        return(res)
    }
    MCEstimator(x = x, ParamFamily = G, criterion = negLoglikelihood)
@
The user can specialize the behavior of \code{MCEstimator} on two layers:
instance-individual or class-individual.

Using the first layer, we may specify model-individual starting values / 
search intervals by
slot \code{startPar} of class \code{ParamFamily}, pass on special control
parameters to functions \code{optim} / \code{optimize} by a \code{...} argument
in function \code{MCEstimator}, and we may enforce valid parameter values
by specifying function slot \code{makeOKPar} of class \code{ParamFamily};
also one can specify a penalty value penalizing invalid parameter values.
E.g.; in case of the censored Poisson distribution family in demo 
{\tt censoredPois} to this package these functions are defined as
<<censPoisFamilyDef, eval = FALSE>>=
    ## search interval for reasonable parameters
    startPar <- function(x,...) c(.Machine$double.eps,max(x))

    ## what to do in case of leaving the parameter domain
    makeOKPar <- function(param) {if(param<=0) return(.Machine$double.eps)
                                  return(param)}
@

In some situations, one would rather like to define rules for groups
of models or to be even more flexible; this can be achieved using the 
class-individual layer: We may use method dispatch to find
the ``right'' function to determine the MC estimator; to this end subclasses
to class \code{L2ParamFamily} have to be defined, which has alread been done,
e.g. in case of  class \code{PoisFamily}. In general these sub classes will not
have any new slots. E.g.; the code to define class \code{PoisFamily} simply is

<<PoisFamilyDef, eval = FALSE>>=
setClass("PoisFamily", contains = "L2ParamFamily")
@

For group models, like the location scale model, there may be additional
slots and intermediate classes. E.g.,
<<NormLocationFamily, eval = FALSE>>=
setClass("NormLocationFamily", contains = "L2LocationFamily")
@

Then, for these subclasses, particular methods may be defined; so far,
in package \pkg{distrMod} we have particular \code{validParameter} methods
for classes \code{ParamFamily}, \code{L2ScaleFamily},
\code{L2LocationFamily}, and \code{L2LocationScaleFamily}.
E.g.; the code to signature \code{L2ScaleFamily} simply is 
<<L2ScaleFamily, eval = FALSE>>=
 setMethod("validParameter", signature(object = "L2ScaleFamily"),
          function(object, param, tol=.Machine$double.eps){
             if(is(param,"ParamFamParameter"))
                param <- main(param)
             if(!all(is.finite(param))) return(FALSE)
             if(length(param)!=1) return(FALSE)
             return(param > tol)})
@

To move the whole model from one parameter value to the other, so far we have
\code{modifyModel} methods for classes  \code{L2ParamFamily}, 
\code{L2LocationFamily}, \code{L2ScaleFamily}, \code{L2LocationScaleFamily},
\code{GammaFamily}, and \code{ExpScaleFamily}, where the second argument to 
dispatch on so for has to be of class \code{ParamFamParameter}.
E.g.; the code to signature \code{model="GammaFamily"} is 
<<GammaFamilyModify, eval = FALSE>>=
setMethod("modifyModel", signature(model = "GammaFamily",
           param = "ParamFamParameter"),
          function(model, param, ...){
             M <- modifyModel(as(model, "L2ParamFamily"), param = param,
                              .withCall = FALSE)
             M@L2derivSymm <- FunSymmList(OddSymmetric(SymmCenter = 
                                                       prod(main(param))),
                                          NonSymmetric())
             class(M) <- class(model)
             return(M)
          })
@ 

We also allow for particular methods within function \code{MCEstimator},
as therein we call method \code{mceCalc}; so far there only is
a method for \code{signature(x="numeric", PFam="ParamFamily")}
Similarly, and more important, the same technique is applied for the wrapper 
function \code{MLEstimator}.

In case of the maximum likelihood estimator as well as in case of minimum
distance (MD) estimation there are the function \code{MLEstimator} and 
\code{MDEstimator} which provide user-friendly interfaces to \code{MCEstimator}.
Hence, the maximum likelihood estimator and for instance the Kolmogorov MD 
estimator can more easily be computed as follows.
<<MLEstimator, eval = TRUE>>=
    MLEstimator(x = x, ParamFamily = G)
    MDEstimator(x = x, ParamFamily = G, distance = KolmogorovDist)
@
Within \code{MLEstimator}, we call method \code{mleCalc}, which then 
dispatches according to its arguments \code{x} and \code{PFam}
as in case of method \code{mceCalc}. So far \code{x} must inherit from
class \code{numeric}, and there are particular methods for argument \code{PFam}
of  classes \code{ParamFamily}, \code{BinomFamily},
\code{PoisFamily}, \code{NormLocationFamily}, \code{NormScaleFamily}, and
\code{NormLocationScaleFamily}. More specifically, \code{mleCalc} must have
an extra \code{...} argument to cope with different callings from \code{MLEstimator};
additional arguments are possible of course. The return value must be 
a list with prescribed structure; to this end function \code{meRes()} is helpful
which produces this structure. 
E.g. the \code{mleCalc}-method for \code{signature(x="numeric", PFam="NormScaleFamily")}
is
<<NormScaleFam, eval = FALSE>>=
setMethod("mleCalc", signature(x = "numeric", PFam = "NormScaleFamily"),
           function(x, PFam, ...){
           n <- length(x)
           theta <- sqrt((n-1)/n)*sd(x); mn <- mean(distribution(PFam))
           ll <- -sum(dnorm(x, mean=mn, sd = theta, log=TRUE))
           names(ll) <- "neg.Loglikelihood"
           crit.fct <- function(sd)
                         -sum(dnorm(x, mean=mn, sd = sd, log=TRUE))  
           param <- ParamFamParameter(name = "scale parameter", 
                               main = c("sd"=theta))
           if(!hasArg(Infos)) Infos <- NULL
           return(meRes(x, theta, ll, param, crit.fct, Infos = Infos))
})
@

We also provide a coercion to class \code{mle} from package \pkg{stats4},
hence making profiling by the \code{profile}-method therein possible.
In order to be able to do so, we need to fill a functional slot 
\code{criterion.fct} of class \code{MCEstimate}. In many examples this
is straightforward, but in higher dimensions, helper function 
\code{get.criterion.fct} can be useful, e.g. it handles the general case
for \code{signature(PFam="ParamFamily")}.


The results of our computations in functions\code{MCEstimator},
\code{MDEstimator},  and \code{MLEstimator} are objects of S4-class \code{MCEstimate}
which inherits from S4-class \code{Estimate}. The definitions are given
in Figure~\ref{figEst}.
\ifpdf
\begin{figure}[!ht]
\vspace{2ex}
  \begin{center}
    \includegraphics[width=4cm]{Estimate.pdf}%
    \caption{\label{figEst}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{Estimate} where we do not
    repeat inherited slots
    }}
  \end{center}
\vspace{-4ex}
\end{figure}
\else
\begin{figure}[htb]
  \begin{center}
    \includegraphics[width=4cm]{Estimate.eps}%
    \caption{\label{figEst}{\footnotesize Inheritance relations and slots of the 
    corresponding \mbox{(sub-)}classes for \code{Estimate} where we do not 
    repeat inherited slots
    }}
  \end{center}
\vspace{-1ex}
\end{figure}
\fi
For  class \code{MCEstimate}, we have a method \code{confint}, which produces
confidence intervals (of class \code{Confint}). 
For class \code{Confint}  as well as for class \code{Estimate} we have
particular \code{show} and \code{print} methods where you may scale the
output by setting global options with \code{distrModOptions}, see also 
subsection~\ref{distrModO}.
As example consider the following:
<<CIex, eval = TRUE, fig = TRUE>>=
require(distrMod)
## some transformation
mtrafo <- function(x){
     nms0 <- c("scale","shape")
     nms <- c("shape","rate")
     fval0 <- c(x[2], 1/x[1])
     names(fval0) <- nms
     mat0 <- matrix( c(0, -1/x[1]^2, 1, 0), nrow = 2, ncol = 2,
                     dimnames = list(nms,nms0))                          
     list(fval = fval0, mat = mat0)}

set.seed(123)
x <- rgamma(50, scale = 0.5, shape = 3)

## parametric family of probability measures
G <- GammaFamily(scale = 1, shape = 2, trafo = mtrafo)
## MLE
res <- MLEstimator(x = x, ParamFamily = G)
print(res, digits = 4, show.details="maximal")
print(res, digits = 4, show.details="medium")
print(res, digits = 4, show.details="minimal")
ci <- confint(res)
print(ci, digits = 4, show.details="maximal")
print(ci, digits = 4, show.details="medium")
print(ci, digits = 4, show.details="minimal")

## some profiling
par(mfrow=c(2,1))
plot(profile(res))
@
% -----------------------------------------------------------------------------
\section{Options}\label{options}
% -----------------------------------------------------------------------------
\subsection[Options for distr]{Options for \pkg{distr}}
% -----------------------------------------------------------------------------
Analogously to the \code{options} command in {\sf R} you may specify a number of
global ``constants'' to be used within the package. These include
\begin{itemize}
  \item \code{DefaultNrFFTGridPointsExponent}: the binary logarithm of the 
        number of grid-points used in FFT ---default $12$
  \item \code{DefaultNrGridPoints}: number of grid-points used for a continuous 
        variable ---default $4096$
  \item \code{DistrResolution}: the finest step length that is permitted for a 
        grid for a discrete variable ---default $1{\rm e-}\!06$
  \item \code{RtoDPQ.e}: For simulational determination of \code{d}, \code{p} 
         and \code{q}, $10^{\rm RtoDPQ.e}$ random variables are
  simulated ---default $5$
  \item \code{TruncQuantile}: to work with compact support, random variables are 
        truncated to their lower/upper
  \code{TruncQuantile}-quantile   ---default $1{\rm e-}\!05$.\\
  From version 1.9 on, for $\varepsilon={\tt \code{TruncQuantile}}$, we use calls of form 
  \code{q(X)(eps, lower.tail = FALSE)} instead of   \code{q(X)(1-eps)} to gain higher
  precision.
  \item \code{warningSim}: controls whether a warning issued at printing/showing 
        a \code{Distribution} object the slots of which have been
  filled starting with simulations ---default \code{TRUE}
  \item \code{warningArith}: controls whether a warning issued at
   printing/showing a \code{Distribution} object produced by arithmetics
  operating on distributions ---default \code{TRUE}
  \item \code{withgaps}: controls whether in the return value of arithmetic
  operations the slot \code{gaps} of an the \code{AbscontDistribution} part
  is filled automatically based on empirical evaluations via  \code{setgaps}
   ---default \code{TRUE}
  \item \code{simplifyD}: controls whether in the return value of arithmetic
  operations there is a call to \code{simplifyD} or not ---default \code{TRUE}
  \item \code{DistrCollapse}: logical;  
  shall support points with distance smaller than \code{DistrResolution} be 
  collapsed; default value:  \code{TRUE}
   \item \code{withSweave}: logical; is code run in Sweave (then no new graphic devices are opened);
                         default value: \code{FALSE}
  \item \code{DistrCollapse.Unique.Warn}: logical;  
  shall a warning be issued upon collapsing? default value:  \code{TRUE}
\item \code{use.generalized.inverse.by.default} %
which is a logical variable 
giving the default value for argument \code{generalized} of our
method \code{solve} in package \pkg{distrMod}. This argument decides whether
our method \code{solve} is to use generalized inverses if the original 
\code{solve}-method from package \pkg{base} fails; if the option is set to 
\code{FALSE}, in case of failure, and unless argument \code{generalized} 
is not explicitely set to \code{TRUE}, \code{solve} will throw an error as is 
the \pkg{base}-method behavior. The default value of thie option is \code{TRUE}.  
%
\end{itemize}
All current options may be inspected by \code{distroptions()}  and modified by
 \linebreak[4] \code{distroptions("<options-name>"=<value>)}.\\
As \code{options},  \code{distroptions("<options-name>")} returns a list of
 length $1$ with the value of the corresponding option,
so here, just as \code{getOption},  \code{getdistrOption("<options-name>")} will
 be preferable, which only returns the value.
%\newline0??????
\subsection[Options for distrEx]{Options for \pkg{distrEx}}\label{distrExoptions}
Up to version {\tt 0.4-4} we used the function
 \code{distrExOptions(arg = "missing", value = -1)} to manage some global 
 options for \pkg{distrEx}, i.e.:\\
\code{distrExOptions()} returns a list of these options,
\code{distrExOptions(arg=x)} returns option \code{x}, and
\code{distrExOptions(arg=x,value=y)} sets the value of option \code{x} to
 {\tt y}.\\
 From version {\tt 1.9} on, we use a mechanism analogue to the
  \code{distroptions}/\code{getdistrOption} commands: 
You may specify certain
global output options to be used within the package with
 \code{distrExoptions}/\code{getdistrExOption}. These include
\begin{itemize}
  \item \code{MCIterations}: number of Monte-Carlo iterations used for crude
          Monte-Carlo integration; defaults to \code{1e5}.
  \item \code{GLIntegrateTruncQuantile}: If \code{integrate} fails and there are
          infinite integration limits, the function \code{GLIntegrate} is
          called inside of \code{distrExIntegrate} with the corresponding
          quantiles  \code{GLIntegrateTruncQuantile} resp.\
          \code{1-GLIntegrateTruncQuantile} as finite integration limits;
          defaults to \code{10*.Machine$double.eps}.
  \item \code{GLIntegrateOrder}: The order used for the Gau{\ss{}}-Legendre
          integration inside of \code{distrExIntegrate}; defaults to 500.
  \item \code{ElowerTruncQuantile}: The lower limit of integration used inside
       of \code{E} which corresponds to the \code{ElowerTruncQuantile}-quantile;
        defaults to \code{1e-7}.
  \item \code{EupperTruncQuantile}: The upper limit of integration used inside 
          of \code{E} which corresponds to the
          \code{(1-ElowerTruncQuantile)}-quantile;
        defaults to \code{1e-7}.
  \item \code{ErelativeTolerance}: The relative tolerance used inside of 
        \code{E} when calling\linebreak[4] \code{distrExIntegrate};
        defaults to \code{.Machine$double.eps^0.25}.
  \item \code{m1dfLowerTruncQuantile}: The lower limit of integration used 
        inside of \code{m1df} which corresponds to the
          \code{m1dfLowerTruncQuantile}-quantile;
        defaults to \code{0}.
  \item \code{m1dfRelativeTolerance}: The relative tolerance used inside of
          \code{m1df} when calling \code{distrExIntegrate};
        defaults to \code{.Machine$double.eps^0.25}.
  \item \code{m2dfLowerTruncQuantile}: The lower limit of integration used 
        inside of \code{m2df} which corresponds to the
          \code{m2dfLowerTruncQuantile}-quantile;
        defaults to \code{0}.
  \item \code{m2dfRelativeTolerance}: The relative tolerance used inside of
          \code{m2df} when calling \code{distrExIntegrate};
        defaults to \code{.Machine$double.eps^0.25}.
 \item \code{nDiscretize}: number of support values used for the discretization
          of objects of class \code{"AbscontDistribution"}; defaults to 100. 
 \item \code{hSmooth}: smoothing parameter to smooth objects of class 
          \code{"DiscreteDistribution"}. This is done via convolution with
          the normal distribution \code{Norm(mean = 0, sd = hSmooth)};
          defaults to \code{0.05}.
 \item \code{IQR.fac}: for determining sensible integration ranges, we use both
          quantile and scale based methods; for the scale based method
          we use the median of the distribution $\pm$ \code{IQR.fac} $\times$ 
          the IQR; defaults to 15.        
\end{itemize}

%We are planning to switch to \code{distroptions}/\code{getdistrOption}-like 
% commands in the next release of this package.
%
\subsection[Options for distrMod]{Options for \pkg{distrMod}}  \label{distrModO}
Just as with to the \code{distroptions}/\code{getdistrOption} commands you 
may specify certain global output options to be used within the package with 
\code{distrModoptions}/\linebreak[4] \code{getdistrModOption}. These include
\begin{itemize}
%
  \item \code{show.details} which controls the detailedness for method 
  \code{show} for objects of classes of the \pkg{distrXXX} family of packages. Possible
  values are
  \begin{itemize}
  \item \code{"maximal"}: {all information is shown}
  \item \code{"minimal"}: {only the most important information is shown}
  \item \code{"medium"}: {somewhere in the middle; see actual \code{show}-methods
  for details.}
  \end{itemize}
    The default value is \code{"maximal"}.
\end{itemize}
%
\subsection[Options for distrSim]{Options for \pkg{distrSim}}
Just as with to the \code{distroptions}/\code{getdistrOption} commands you 
may specify certain global output options to be used within the package with 
\code{distrSimoptions}/\linebreak[4] \code{getdistrSimOption}. These include
\begin{itemize}
  \item \code{MaxNumberofPlottedObs} the maximal number of observation plotted 
  in a plot of an object of class \code{Dataclass}; defaults to 4000

  \item \code{MaxNumberofPlottedObsDims}: the maximum number of observations to 
  be plotted in a plot of an object of class \code{Dataclass}
  and descendants; defaults to 6.
  \item \code{MaxNumberofPlottedRuns}: the maximum number of runs to be plotted 
  in a plot of an object of class \code{Dataclass}
  and descendants (one run/panel); defaults to 6.
  \item \code{MaxNumberofSummarizedObsDims}: the maximum number of observations 
  to be summarized of an object of class \code{Dataclass}
  and descendants; defaults to 6.
  \item \code{MaxNumberofSummarizedRuns}: the maximum number of runs to be
   summarized of an object of class \code{Dataclass}
  and descendants; defaults to 6.
\end{itemize}
%
\subsection[Options for distrTEst]{Options for \pkg{distrTEst}}
Just as with to the \code{distroptions}/\code{getdistrOption} commands you may 
specify certain
global output options to be used within the package with 
\code{distrTEstoptions}/ \linebreak[4]\code{getdistrTEstOption}. These include
\begin{itemize}
  \item \code{MaxNumberofPlottedEvaluations}:  the maximal number of evaluations 
  to be plotted
  in a plot of an object of class \code{EvaluationList}; defaults to 6
  \item \code{MaxNumberofPlottedEvaluationDims}: the maximal number of
   evaluation dimensions to be plotted in a plot of an
         object of class \code{Evaluation}; defaults to 6
  \item \code{MaxNumberofSummarizedEvaluations}: the  maximal number of 
  evaluations to be summarized of an object of class
  \code{EvaluationList}; defaults to 15
  \item \code{MaxNumberofPrintedEvaluations}: the maximal number of evaluations
   printed of an object of class
  \code{EvaluationList}; defaults to 15
\end{itemize}

%??????1

\section{Further Documentation}
\subsection{Help pages}
Additional information can be obtained during an {\sf R} session,
using help files and startup messages.
\paragraph{Startup messages}
Upon loading packages from the {\tt distrXXX} family of packages 
with \code{require} or \code{library},
by default startup messages pop up giving some starting points where
to look for further information; howto scale/switch off these
startup messages will be discussed in the next section.
\paragraph{Package help file}
A starting point is doubtless the package help files,
called e.g.\ by \code{?distr}. We give an {\tt ASCII} form
of a class graph with all the {\tt S4} classes in the 
corresponding package, and indicate the implemented methods and functions.
From version 2.1 this help file for \pkg{distr} has a  
new section "Extension packages" pointing to the various extensions
of this package. 
\paragraph{Individual help files}
Of course, for looking up the syntax of a function the 
help pages available with \code{?} / \code{help} are extremely
useful. Note that for our packages, help pages are also available
for internal (i.e.; non exported) functions.
\subsection[NEWS file]{{\tt NEWS} file}
To get more details on the implementation process you may also
consult a corresponding {\tt NEWS} file where all relevant changes 
are listed at the package version where they have been implemented.
You may inspect this {\tt NEWS} file with \code{NEWS(<pkgname>)}.
\subsection{Vignettes}
Besides this large (or huge) vignette, there are also some minor
vignettes available, assembling documentation somewhat more coherent
than in the help pages. From version 2.1 on, e.g., there is a new vignette
``How to generate new distributions in packages distr, distrEx''
in package \pkg{distr} which
is to encourage the implementation of new distributions  and 
distribution classes by third parties.
\subsection{Articles}
Of course we plan to publish some of our findings in peer-reviewed journals;
so the part of this vignette dealing with package \pkg{distrMod} will
form the basis for an article on this package by the present authors.

\section{Startup Messages}
%
For the management of startup messages, from version~1.7, we use package 
\pkg{startupmsg}:
When loading/attaching packages \pkg{distr}, \pkg{distrEx}, \pkg{distrSim}, or 
\pkg{distrTEst} for each package a disclaimer is displayed.

You may suppress these start-up banners/messages completely by setting 
\linebreak[4]\code{options("StartupBanner"="off")}
somewhere before loading this package by \code{library} or \code{require} in 
your R-code / R-session.

If option \code{"StartupBanner"} is not defined (default) or setting\linebreak[4]
\code{options("StartupBanner" = NULL)} or %\linebreak[4]
\code{options("StartupBanner" = "complete")}
the complete start-up banner is displayed.

For any other value of option \code{"StartupBanner"} (i.e., not in 
\linebreak[3]\code{c(NULL, "off", "complete")})
only the version information is displayed.

The same can be achieved by wrapping the \code{library} or \code{require}  
call into either
 \linebreak[4]\code{onlytypeStartupMessages(<code>, atypes="version")} 
 or\linebreak[4] \code{suppressStartupMessages(<code>)}.
%

\section[System/version requirements]{System/version requirements, license,
 etc.\ }
%
\subsection{System requirements}
%
As our package is completely written in {\sf R}, there are no dependencies
on the underlying OS; of course, there is the usual speed gain
possible on recent machines. We have tested our package on a Pentium~II with
 233 MHz, on Pentium~III's with 0.8--2.1 GHz, and on an Athlon with 2.5 GHz 
 giving a reasonable performance.

\subsection{Required version of {\sf R}}
Contrary to the hardware required,
if you want to use \code{library} or \code{require} to use  \pkg{distr} 
within {\sf R} code, you need at least {\sf R} Version {\tt 1.8.1},
as we make use of name space operations only available from that version on;
also, the command \code{setClassUnion}, which is used in some sources, is only
 available from that version on.\\
%
On the other hand, if the package may be pasted in by \code{source}, the code 
works with {\sf R} from
version {\tt 1.7.0} on ---but without using name-spaces, which is only available
 from {\tt 1.8.0} on.
Due to some changes in {\sf R} from version {\tt 1.8.1} to {\tt 1.9.0} and from 
{\tt 1.9.1} to {\tt 2.0.0}, we have to provide different
 {\tt zip}/{\tt tar.gz}-Files for these versions.\\
Versions of \pkg{distr} from version number {\tt 1.5} onwards are only
supplied for {\sf R} Version {\tt 2.0.1 patched} and later.
After a reorganization, versions of \pkg{distr} from version number {\tt 1.6} 
onwards are only supplied for {\sf R} Version {\tt 2.2.0 patched} and later.


\subsection{Dependencies}
In package \pkg{distr}, from version 2.0, we make use of \code{D1ss} from
\href{mailto:maechler@stat.math.ethz.ch}{Martin M\"achler's} package \pkg{sfsmisc}.
In package \pkg{distrEx}, we need 
\href{mailto:alec_stephenson@hotmail.com}{Alec Stephenson's} package \pkg{evd} 
for the extreme  value distributions implemented therein, as well as 
\href{mailto:Vincent.Goulet@act.ulaval.ca}{Vincent Goulet}'s and Mathieu Pigeon's
package \pkg{actuar} for the (single parameter) Pareto distribution (from
\pkg{distrEx} version 2.1 on) which has been ported to our framework by
\href{mailto:Nataliya.Horbenko@itwm.fraunhofer.de}{Nataliya Horbenko}. 

In package \pkg{distrSim}, and conseqently also in package \pkg{distrTEst} we 
use \href{mailto:pgilbert@bank-banque-canada.ca}{Paul Gilbert's}  
package \pkg{setRNG}
to be installed from \href{http://cran.r-project.org/mirrors.html}{\tt CRAN}
for the control of the seed of the random number generator in our simulation 
classes.
More precisely, for our version $\le$ {\tt 1.6}  we need his
 version $<$ {\tt 2006.2-1},
and for our version $\ge$ {\tt 1.7} we need his version $\ge$ {\tt 2006.2-1}.

From package version {\tt 1.7}/{\tt 0.4-3} on, we also need package
 \pkg{startupmsg} by the first of the present authors, which also is available 
 on \href{http://cran.r-project.org/mirrors.html}{\tt CRAN}.

\subsection{License}
This software is distributed under the terms of the GNU GENERAL
PUBLIC LICENSE Version 2, June 1991, more specifically under {\tt LGPL-3}
confer\newline
%\href{http://www.gnu.org/copyleft/gpl.html}%
%{\tt http://www.gnu.org/copyleft/gpl.html}
\href{http://www.gnu.org/copyleft/gpl.html}%
{\footnotesize \tt http://www.gnu.org/copyleft/gpl.html},
\href{http://en.wikipedia.org/wiki/GNU_Lesser_General_Public_License}%
{\footnotesize\tt \url{http://en.wikipedia.org/wiki/GNU_Lesser_General_Public_License}}

\section{Details to the implementation}
\begin{itemize}
\item As the normal distribution is closed under affine transformations, we have 
overloaded the corresponding methods. 
%The same goes for the exponential distribution under scale transformations.
\item For the general convolution algorithm for univariate probability 
distribution functions/densities by means of FFT, which we use in the 
overloaded {\tt "+"}-operator, confer~\cite{K:R:S:04}.
\item Exact convolution methods are implemented for the normal, the Poisson, 
the binomial, the negative binomial, the Gamma (and the \code{Exp}), and 
the {$\chi^2$} distribution
\item Exact formulae for scale transformations are implemented for 
the Exp-/Gamma-distribution, the Weibull and the log-normal distribution
(the latter two from version 1.9 on). 
\item Exact formulae for affine linear transformations are available for
the normal, the logistic and the Cauchy distribution (the latter two from
version 1.9 on).  
\item Instances of any class transparent to the user are initialized 
by\linebreak[4] {\tt<classname>([<slotname>=<value>,...])}
where except for class \code{DataClass} in package \pkg{distrSim} all classes 
have default values for all their slots; in \code{DataClass},
the slot \code{Data} has to be specified.
\item Multiplication (and Division) is implemented as corresponding
exponentials of the convolution of the logarithms (evaluated separately
for positive and negative parts).
\item Exponentiation also uses the exp-log trick.
\item Multiplication, Exponentation, and Min/Maximum of an \code{AbscontDistribution}
and a \code{DiscreteDistribution} as an intermediate step produce
a \code{UnivarMixingDistribution}, with one mixing component for each element
of the support of the \code{DiscreteDistribution}. As a last step,
this \code{UnivarMixingDistribution} is then ``flattened''.
\item
As suggested in \cite{OOPGent} all slots are accessed and modified by 
corresponding accessor- and replacement functions ---templates for which were 
produced by \code{standardMethods}.

{\bf We strongly discourage the use of the \code{@}-operator to modify
or even access slots \code{r}, \code{d}, \code{p}, and \code{q}, confer
Example~{\ref{destrex}}.}
\end{itemize}
%
\section{A general utility}
%
Following \cite{OOPGent}, the programmer of {\tt S4}-classes should provide
accessor and replacement functions for the inspection/modification of
any newly introduced slot. This can be quite a task when you have a 
lot of classes/slots.
As these functions all have the same structure, it would be nice
to automatically generate templates for them.
Faced with this problem in developing this package, Thomas Stabla
has written such a utility, \code{standardMethods} ---which the authors
of this package recommend for any developer of {\tt S4}-classes.
For more details, see \code{?standardMethods}.

\section{Odds and Ends}
\subsection[What should be done and what we could do]%
{What should be done and what we could do ---for version $>$\pkgversion}
\begin{itemize}
\item application of analytic FourierTransforms instead of FFT where
appropriate ---perhaps also to be controlled
by a parameter/option
\item use the \code{q}-slot applied to \code{runif} in \code{simplifyr} 
for continuous distributions
\item further exact formulae for binary arithmetic operations like \code{"*"}
%\item derivation of a class \code{LatticeDistribution} from 
%\code{DiscreteDistribution} to be able to easily apply FFT
%\item redo the math-method for discrete distributions when only slot \code{r} %
%is given
%\item better documentation for method \code{evaluate} / class \code{Evaluation}
%\item adaptation of class \code{Evaluation} / method \code{evaluate} to be more flexible
%\item use of  \code{initialize} in generating functions
%\item generating function for new distribution classes to ease inclusion of 
%new distributions
\item goodness of fit tests for distribution-objects
%\item use of $\tt \backslash$\code{S4method} in documentation
%\item overloading binary operators of group \code{Math2} for independent 
%distributions
\item defining a subgroup of \code{Math2} of invertible binary operators
%%\item better use of \code{concept} in rd-files
%\item suggestion by \href{mailto:kouros.owzar@duke.edu}{Kouros Owzar}: 
%in case of parameters of dimension $p>1$
%---as in case of ${\cal N}(\mu,\sigma^2)$--- possibility to %
%access/replace that parameter  as a vector
\end{itemize}
\subsection{What should be done but for which we lack the know-how}
\begin{itemize}
  \item multivariate distributions
  \item conditional distributions
  \item copula
\end{itemize}
%
\section{Acknowledgement}
In order to give our acknowledgements their due place
in the manual, we insert them before some rather
extensive presentation of examples, because otherwise
they would probably get lost or overseen by most of the
readers.

We thank Martin M\"achler and Josef Leydold for their
helpful suggestions in conceiving the package.
John Chambers also gave several helpful hints and
insights when responding to our requests concerning the
{\tt S4}-class concept in {\tt r-devel}/ {\tt r-help}.
We got stimulating replies to an RFC on {\tt r-devel} by
Duncan Murdoch and Gregory Warnes.
We also thank Paul Gilbert for drawing our attention to his package
 {\tt setRNG} and making it available as stand-alone version.
In the last few days before the release on {\tt CRAN}, Kurt Hornik and
Uwe Ligges were very kind, helping us to find the clue how to
pass all necessary checks by {\tt R CMD check}.
We also thank G. Jay Kerns for contributing code for
the skewness and kurtosis functionals.

Last not least a big "thank you" to Torsten Hothorn as editor
of {\tt R-News}, for his patience with our endless versions
until we finally came to a publishable version.
%\input{distrack}
%
\section{Examples}
\subsection{$12$-fold convolution of uniform $(0,1)$ variables}
\begin{footnotesize}
  Code also available under\newline %
  {\href{http://www.uni-bayreuth.de/departments/math/org/mathe7/DISTR/NormApprox.R}%
  {\parbox[t]{12cm}{$\mbox{\hspace{2cm}}$%
  {\tt http://www.uni-bayreuth.de/departments/math/org/}\\%
  {$\mbox{\hspace{2cm}}$\hphantom{\tt http:/}%
  {\tt /mathe7/DISTR/NormApprox.R}}}}}\\[2ex]
\end{footnotesize}
\begin{small}
  This example shows how easily we may get the distribution of the sum of $12$
  ${\rm i.i.d.}$ ${\rm ufo}(0,1)$--variables minus $6$--- which was used as a 
  fast generator of ${\cal N}(0,1)$--variables in times when evaluations of 
  $\exp$, $\log$, $\sin$ and $\tan$ were expensive, confer
  \cite{Ric:88}, example~C, p.~163. The user should not be confused by
   expressions  like {\tt U+U}: this {\em does not\/} mean {\tt 2U}
  but rather convolution of two independent identically distributed 
  random variables.
\end{small}
<<NormApprox, eval = TRUE, fig = TRUE>>=
require(distr)

N <- Norm(0,1)
U <- Unif(0,1)
U2 <- U + U 
U4 <- U2 + U2
U8 <- U4 + U4
U12 <- U4 + U8
NormApprox <- U12 - 6

x <- seq(-4,4,0.001)

opar <- par(no.readonly = TRUE)
par(mfrow = c(2,1))

plot(x, d(NormApprox)(x),
     type = "l",
     xlab = "",
     ylab = "Density",
     main = "Exact and approximated density")
lines(x, d(N)(x),
      col = "red")
legend("topleft",
       legend = c("NormApprox", "Norm(0,1)"),
       fill = c("black", "red"))

plot(x, d(NormApprox)(x) - d(N)(x),
     type = "l",
     xlab = "",
     ylab = "\"black\" - \"red\"",
     col = "darkgreen",
     main = "Error")
lines(c(-4,4), c(0,0))

par(opar)
@
\subsection{Comparison of exact convolution to FFT for normal distributions}\label{Convex}
\begin{footnotesize}
  Code also available under\newline \href{http://www.uni-bayreuth.de/departments/math/org/mathe7/DISTR/ConvolutionNormalDistr.R}%
  {\parbox[t]{12cm}{$\mbox{\hspace{2cm}}${\tt http://www.uni-bayreuth.de/departments/math/org/}\\%
  {$\mbox{\hspace{2cm}}$\hphantom{\tt http:/}{\tt /mathe7/DISTR/ConvolutionNormalDistr.R}}}}\\[2ex]
\end{footnotesize}
\begin{small}
  This example illustrates the exactness of the numerical algorithm used to 
  compute the convolution: We know that ${\cal L}({\tt A+B})={\cal N}(5,13)$ --- 
  if the second argument of ${\cal N}$ is the variance
\end{small}
<<ConvolutionNormalDistr, eval = TRUE, fig = TRUE>>=
require(distr)

## initialize two normal distributions
A <- Norm(mean=1, sd=2)
B <- Norm(mean=4, sd=3) 

## convolution via addition of moments
AB <- A+B

## casting of A,B as absolutely continuous distributions
## that is, ``forget'' that A,B are normal distributions
A1 <- as(A, "AbscontDistribution")
B1 <- as(B, "AbscontDistribution")

## for higher precision we change the global variable
## "TruncQuantile" from 1e-5 to 1e-8
oldeps <- getdistrOption("TruncQuantile")
eps <- 1e-8
distroptions("TruncQuantile" = eps)
## support of A1+B1 for FFT convolution is
## [q(A1)(TruncQuantile), 
##  q(B1)(TruncQuantile, lower.tail = FALSE)]

## convolution via FFT
AB1 <- A1+B1

#############################
## plots of the results
#############################
par(mfrow=c(1,3))
low <- q(AB)(1e-15)
upp <- q(AB)(1e-15, lower.tail = FALSE)
x <- seq(from = low, to = upp, length = 10000)

## densities
plot(x, d(AB)(x), type = "l", lwd = 5)
lines(x , d(AB1)(x), col = "orange", lwd = 1)
title("Densities")
legend("topleft", legend=c("exact", "FFT"), 
        fill=c("black", "orange"))

## cdfs
plot(x, p(AB)(x), type = "l", lwd = 5)
lines(x , p(AB1)(x), col = "orange", lwd = 1)
title("CDFs")
legend("topleft", legend=c("exact", "FFT"), 
        fill=c("black", "orange"))

## quantile functions
x <- seq(from = eps, to = 1-eps, length = 1000)
plot(x, q(AB)(x), type = "l", lwd = 5)
lines(x , q(AB1)(x), col = "orange", lwd = 1) 
title("Quantile functions")
legend("topleft", legend=c("exact", "FFT"), 
        fill=c("black", "orange"))

## Since the plots of the results show no 
## recognizable differencies, we also compute 
## the total variation distance of the densities 
## and the Kolmogorov distance of the cdfs

## total variation distance of densities
total.var <- function(z, N1, N2){
    0.5*abs(d(N1)(z) - d(N2)(z))
}
dv <- integrate(total.var, lower=-Inf, upper=Inf, rel.tol=1e-8, N1=AB, N2=AB1)
cat("Total variation distance of densities:\t")
print(dv) # 4.25e-07

### meanwhile realized in package "distrEx" 
### as TotalVarDist(N1,N2)

## Kolmogorov distance of cdfs 
## the distance is evaluated on a random grid
z <- r(Unif(Min=low, Max=upp))(1e5)
dk <- max(abs(p(AB)(z)-p(AB1)(z)))
cat("Kolmogorov distance of cdfs:\t", dk, "\n") 
# 2.03e-07

### meanwhile realized in package "distrEx" 
### as KolmogorovDist(N1,N2)

## old distroptions
distroptions("TruncQuantile" = oldeps)

@
\subsection{Comparison of FFT to {\tt RtoDPQ}}\label{compex}
\begin{footnotesize}
  Code also available under\newline \href{http://www.uni-bayreuth.de/departments/math/org/mathe7/DISTR/ComparisonFFTandRtoDPQ.R}%
  {\parbox[t]{12cm}{$\mbox{\hspace{2cm}}${\tt http://www.uni-bayreuth.de/departments/math/org/}\\%
  {$\mbox{\hspace{2cm}}$\hphantom{\tt http:/}{\tt /mathe7/DISTR/ComparisonFFTandRtoDPQ.R}}}}\\[2ex]
\end{footnotesize}
\begin{small}
  This example illustrates the exactness (or rather not--so--exactness) of the 
  simulational default algorithm used to compute the distribution of 
  transformations of group {\tt math}.
\end{small}
<<ComparisonFFTandRtoDPQ.R, eval = TRUE, fig = TRUE>>=
require(distr)

################################
## Comparison 1 - FFT and RtoDPQ 
################################

N1 <- Norm(0,3)
N2 <- Norm(0,4)
rnew1 <- function(n) r(N1)(n) + r(N2)(n) 

X <- N1 + N2 
     # exact formula -> N(0,5)
Y <- N1 + as(N2, "AbscontDistribution") 
     # appoximated with FFT
Z <- new("AbscontDistribution", r = rnew1) 
     # appoximated with RtoDPQ

# density-plot

x <- seq(-15,15,0.01)
plot(x, d(X)(x),
     type = "l",
     lwd = 3,
     xlab = "",
     ylab = "density",
     main = "Comparison 1",     
     col = "black")
lines(x, d(Y)(x),
     col = "yellow")
lines(x, d(Z)(x),
     col = "red")
legend("topleft",
  legend = c("Exact", "FFT-Approximation", 
             "RtoDPQ-Approximation"),
       fill = c("black", "yellow", "red"))
      
############################################
## Comparison 2 - "Exact" Formula and RtoDPQ
############################################

B <- Binom(size = 6, prob = 0.5) * 10
N <- Norm()
rnew2 <- function(n) r(B)(n) + r(N)(n)

Y <- B + N 
     # "exact" formula
Z <- new("AbscontDistribution", r = rnew2) 
     # appoximated with RtoDPQ

# density-plot

x  <- seq(-5,65,0.01)
plot(x, d(Y)(x),
     type = "l",
     xlab = "",
     ylab = "density",
     main = "Comparison 2",
     col = "black")
lines(x, d(Z)(x),
     col = "red")
legend("topleft",
       legend = c("Exact", "RtoDQP-Approximation"),
       fill = c("black", "red"))
@
\subsection{Comparison of exact and approximate stationary regressor distribution}\label{statex}
\begin{footnotesize}
  Code also available under\newline \href{http://www.uni-bayreuth.de/departments/math/org/mathe7/DISTR/StationaryRegressorDistr.R}%
  {\parbox[t]{12cm}{$\mbox{\hspace{2cm}}${\tt http://www.uni-bayreuth.de/departments/math/org/}\\%
  {$\mbox{\hspace{2cm}}$\hphantom{\tt http:/}{\tt /mathe7/DISTR/StationaryRegressorDistr.R}}}}\\[2ex]
\end{footnotesize}
\begin{small}
  Another illustration for the use of package {\tt "distr"}.
  In case of a stationary AR(1)--model, for non--normal innovation distribution, 
  the stationary distribution of the observations must be approximated by finite 
  convolutions. That these approximations give fairly good results for
  approximations down to small orders is exemplified by the Gaussian case where 
  we may compare the approximation to the exact stationary distribution.
\end{small}
<<StationaryRegressorDistr, eval = TRUE, fig = TRUE>>=
require(distr)

## Approximation of the stationary regressor 
## distribution of an AR(1) process 
##       X_t = phi X_{t-1} + V_t 
## where V_t i.i.d N(0,1) and phi\in(0,1)
## We obtain 
##    X_t = \sum_{j=1}^\infty phi^j V_{t-j}
## i.e., X_t \sim N(0,1/(1-phi^2))
phi <- 0.5

## casting of V as absolutely continuous distributions
## that is, ``forget'' that V is a normal distribution
V <- as(Norm(), "AbscontDistribution")

## for higher precision we change the global variable
## "TruncQuantile" from 1e-5 to 1e-8
oldeps <- getdistrOption("TruncQuantile")
eps <- 1e-8
distroptions("TruncQuantile" = eps)

## Computation of the approximation 
##      H=\sum_{j=1}^n phi^j V_{t-j}
## of the stationary regressor distribution 
## (via convolution using FFT)
H <- V
n <- 15 
## may take some time
### switch off warnings [would be issued due to 
###  very unequal variances...]
old.warn <- getOption("warn")
options("warn" = -1)
for(i in 1:n){Vi <- phi^i*V; H <- H + Vi } 
options("warn" = old.warn)

## the stationary regressor distribution (exact)
X <- Norm(sd=sqrt(1/(1-phi^2)))

#############################
## plots of the results
#############################
par(mfrow=c(1,3))
low <- q(X)(1e-15)
upp <- q(X)(1e-15, lower.tail = FALSE)
x <- seq(from = low, to = upp, length = 10000)

## densities
plot(x, d(X)(x),type = "l", lwd = 5)
lines(x , d(H)(x), col = "orange", lwd = 1)
title("Densities")
legend("topleft", legend=c("exact", "FFT"), 
        fill=c("black", "orange"))

## cdfs
plot(x, p(X)(x),type = "l", lwd = 5)
lines(x , p(H)(x), col = "orange", lwd = 1)
title("CDFs")
legend("topleft", legend=c("exact", "FFT"), 
        fill=c("black", "orange"))

## quantile functions
x <- seq(from = eps, to = 1-eps, length = 1000)
plot(x, q(X)(x),type = "l", lwd = 5)
lines(x , q(H)(x), col = "orange", lwd = 1)
title("Quantile functions")
legend( "topleft", 
        legend=c("exact", "FFT"), 
        fill=c("black", "orange"))

## Since the plots of the results show no 
## recognizable differencies, we also compute 
## the total variation distance of the densities 
## and the Kolmogorov distance of the cdfs

## total variation distance of densities
total.var <- function(z, N1, N2){
    0.5*abs(d(N1)(z) - d(N2)(z))
}
dv <- integrate(f = total.var, lower = -Inf, 
                upper = Inf, rel.tol = 1e-7, 
                N1=X, N2=H)
cat("Total variation distance of densities:\t")
print(dv) # ~ 5.0e-06

### meanwhile realized in package "distrEx" 
### as TotalVarDist(N1,N2)


## Kolmogorov distance of cdfs 
## the distance is evaluated on a random grid
z <- r(Unif(Min=low, Max=upp))(1e5)
dk <- max(abs(p(X)(z)-p(H)(z)))
cat("Kolmogorov distance of cdfs:\t", dk, "\n") 
# ~2.5e-06

### meanwhile realized in package "distrEx" 
### as KolmogorovDist(N1,N2)


## old distroptions
distroptions("TruncQuantile" = oldeps)

@
\subsection{Truncation and Huberization/winsorization}\label{truncex}
has been integrated to the package itself, see section~\ref{TruncMin}
\subsection{Distribution of minimum and maximum of two independent random variables}\label{minmaxex}
has been integrated to the package itself, see section~\ref{TruncMin}
\subsection{Instructive destructive example}\label{destrex}
\begin{footnotesize}
  Code also available under\newline \href{http://www.uni-bayreuth.de/departments/math/org/mathe7/DISTR/destructive.R}%
  {\parbox[t]{12cm}{$\mbox{\hspace{2cm}}${\tt http://www.uni-bayreuth.de/departments/math/org/}\\%
  {$\mbox{\hspace{2cm}}$\hphantom{\tt http:/}{\tt /mathe7/DISTR/destructive.R}}}}\\[2ex]
\end{footnotesize}
<<destructive, eval = TRUE, fig = TRUE>>=
##########################################################
## Demo: Instructive destructive example
##########################################################
require(distr)

## package "distr" encourages 
## consistency but does not 
## enforce it---so in general  
## d o   n o t   m o d i f y
## slots d,p,q,r!

N <- Norm()
B <- Binom()
N@d <- B@d
plot(N, lwd = 3) 
@
\subsection{A simulation example}\label{simex}
{\bf needs packages \pkg{distrSim}/\pkg{distrTEst}}\\[2ex]
\begin{footnotesize}
  Code also available under\newline \href{http://www.uni-bayreuth.de/departments/math/org/mathe7/DISTR/SimulateandEstimate.R}%
  {\parbox[t]{12cm}{$\mbox{\hspace{2cm}}${\tt http://www.uni-bayreuth.de/departments/math/org/}\\%
  {$\mbox{\hspace{2cm}}$\hphantom{\tt http:/}{\tt /mathe7/DISTR/SimulateandEstimate.R}}}}\\[2ex]
\end{footnotesize}
<<SimulateandEstimate, eval = TRUE, fig = TRUE, width=8.8,height=3.8>>=
require(distrTEst)
    ### also loads distrSim
sim <- new("Simulation",
           seed = setRNG(),
           distribution = Norm(mean = 0, sd = 1),
           filename="sim_01",
           runs = 1000,
           samplesize = 30)

contsim <- new("Contsimulation",
               seed = setRNG(),
               distribution.id = Norm(mean = 0, sd = 1),
               distribution.c = Norm(mean = 0, sd = 9),
               rate = 0.1,
               filename="contsim_01",
               runs = 1000,
               samplesize = 30)

simulate(sim)
simulate(contsim)

sim
summary(contsim)
plot(contsim)
@
<<elist, eval = TRUE, fig = TRUE>>=
require(distrTEst)
psim <- function(theta,y,m0){
  mean(pmin(pmax(-m0, y - theta), m0))
  }
mestimator <- function(x, m = 0.7) {
  uniroot(f = psim,
          lower = -20,
          upper = 20,
          tol = 1e-10,
          y = x,
          m0 = m,
          maxiter = 20)$root
  }

result.id.mean <- evaluate(sim, mean)
result.id.mest <- evaluate(sim, mestimator)
result.id.median <- evaluate(sim, median)


result.cont.mean <- evaluate(contsim, mean)
result.cont.mest <- evaluate(contsim, mestimator)
result.cont.median <- evaluate(contsim, median)

elist <- EvaluationList(result.cont.mean,
                        result.cont.mest,
                        result.cont.median) 

elist
summary(elist)
plot(elist, cex = 0.7, las = 2)
@
\par
\begin{footnotesize}
Output by \code{plot}/\code{show}-method for an object of class \code{Evaluation}
<<mest, eval = TRUE>>=
result.cont.mest
@
\end{footnotesize}
\begin{footnotesize}
Output by \code{summary}-method for an object of class \code{EvaluationList}
<<elist.summary, eval = TRUE>>=
summary(elist)
@
\end{footnotesize}
\begin{small}
In this example we present a standard robust simulation study that --- in variations --- arises in almost
every paper on Robust Statistics. We do this with the tools provided by our package\ldots
\end{small}
\subsection{Expectation of a given function under a given distribution}
\begin{footnotesize}
  Code also available under\newline \href{http://www.uni-bayreuth.de/departments/math/org/mathe7/DISTR/Expectation.R}%
  {\parbox[t]{12cm}{$\mbox{\hspace{2cm}}${\tt http://www.uni-bayreuth.de/departments/math/org/}\\%
  {$\mbox{\hspace{2cm}}$\hphantom{\tt http:/}{\tt /mathe7/DISTR/Expectation.R}}}}\\[2ex]
  This code is for illustration only; in the mean-time, the expectation- and 
  variance operators implemented in this example have been included to package
   \pkg{distrEx} where their functionality has further been extended.
\end{footnotesize}
\begin{small}
As in examples~\ref{truncex} and \ref{minmaxex}, we illustrate the use of 
package {\tt "distr"} by implementing a general evaluation of expectation 
and variance under a given distribution.
\end{small}
<<Expectation, eval = TRUE>>=
require("distrEx")
# Example
id <- function(x) x
sq <- function(x) x^2

# Expectation and Variance of Binom(6,0.5)
B <- Binom(6, 0.5)
E(B, id)
E(B, sq) - E(B, id)^2

# Expectation and Variance of Norm(1,1)
N <- Norm(1, 1)
E(N, id)
E(N, sq) - E(N, id)^2
@
\subsection{$n$-fold convolution of absolutely continuous distributions}\label{exe10}
\begin{footnotesize}
  Code also available under\newline \href{http://www.uni-bayreuth.de/departments/math/org/mathe7/DISTR/nFoldConvolution.R}%
  {\parbox[t]{12cm}{$\mbox{\hspace{2cm}}${\tt http://www.uni-bayreuth.de/departments/math/org/}\\%
  {$\mbox{\hspace{2cm}}$\hphantom{\tt http:/}{\tt /mathe7/DISTR/nFoldConvolution.R}}}}\\[2ex]
\end{footnotesize}
\begin{small}
Might be useful for teaching the CLT: a straightforward implementation of the 
$n$--fold convolution of an
arbitrary implemented absolutely continuous distribution --- to show accuracy 
of our method we compare it to the
exact formula valid for $n$-fold convolution of normal distributions.\\
From version 1.9 this is integrated to package \pkg{distr}.
\end{small}
<<nFoldConvolution, eval = TRUE, fig = TRUE>>=
##########################################################
## Demo: n-fold convolution of absolutely continuous 
##       probability distributions
##########################################################
require(distr)

if(!isGeneric("convpow")) 
    setGeneric("convpow", 
    function(D1,...) standardGeneric("convpow"))

##########################################################
## Function for n-fold convolution
## -- absolute continuous distribution -- 
##########################################################

##implentation of Algorithm 3.4. of 
# Kohl, M., Ruckdeschel, P., Stabla, T. (2005): 
#   General purpose convolution algorithm for distributions 
#   in S4-Classes by means of FFT.
# Technical report, Feb. 2005. Also available in
# http://www.uni-bayreuth.de/departments/math/org/mathe7/
#       /RUCKDESCHEL/pubs/comp.pdf


setMethod("convpow",
          signature(D1 = "AbscontDistribution"),
          function(D1, N){
            if((N < 1)||(!identical(floor(N), N)))
              stop("N has to be a natural greater than 0")
            
            m <- getdistrOption("DefaultNrFFTGridPointsExponent")

    ##STEP 1

            lower <- ifelse((q(D1)(0) > - Inf), q(D1)(0), 
                     q(D1)(getdistrOption("TruncQuantile"))) 
            upper <- ifelse((q(D1)(1) < Inf), q(D1)(1), 
                     q(D1)(getdistrOption("TruncQuantile"), lower.tail = FALSE))

    ##STEP 2

            M <- 2^m
            h <- (upper-lower)/M
            if(h > 0.01)
              warning(paste("Grid for approxfun too wide, ", 
              "increase DefaultNrFFTGridPointsExponent", sep=""))
            x <- seq(from = lower, to = upper, by = h)
            p1 <- p(D1)(x)

    ##STEP 3

            p1 <- p1[2:(M + 1)] - p1[1:M]

    ##STEP 4
    
            ## computation of DFT
            pn <- c(p1, numeric((N-1)*M))
            fftpn <- fft(pn)

    ##STEP 5

            ## convolution theorem for DFTs
            pn <- Re(fft(fftpn^N, inverse = TRUE)) / (N*M)
            pn <- (abs(pn) >= .Machine$double.eps)*pn
            i.max <- N*M-(N-2)
            pn <- c(0,pn[1:i.max])
            dn <- pn / h
            pn <- cumsum(pn)

    ##STEP 6(density)

            ## density 
            x <- c(N*lower,seq(from = N*lower+N/2*h, 
                   to = N*upper-N/2*h, by=h),N*upper)
            dnfun1 <- approxfun(x = x, y = dn, yleft = 0, yright = 0)

    ##STEP 7(density)
 
            standardizer <- sum(dn[2:i.max]) + (dn[1]+dn[i.max+1]) / 2
            dnfun2 <- function(x) dnfun1(x) / standardizer

    ##STEP 6(cdf)
    
            ## cdf with continuity correction h/2
            pnfun1 <- approxfun(x = x+0.5*h, y = pn, 
                        yleft = 0, yright = pn[i.max+1])

    ##STEP 7(cdf)
   
            pnfun2 <- function(x) pnfun1(x) / pn[i.max+1]


            ## quantile with continuity correction h/2
            yleft <- ifelse(((q(D1)(0) == -Inf)|
                             (q(D1)(0) == -Inf)), 
                             -Inf, N*lower)
            yright <- ifelse(((q(D1)(1) == Inf)|
                              (q(D1)(1) == Inf)), 
                              Inf, N*upper)    
            w0 <- options("warn")
            options(warn = -1)
            qnfun1 <- approxfun(x = pnfun2(x+0.5*h), 
                        y = x+0.5*h, yleft = yleft, yright = yright)
            qnfun2 <- function(x){ 
            ind1 <- (x == 0)*(1:length(x))
            ind2 <- (x == 1)*(1:length(x))
            y <- qnfun1(x)
            y <- replace(y, ind1[ind1 != 0], yleft)
            y <- replace(y, ind2[ind2 != 0], yright)
            return(y)
            }
            options(w0)

            rnew = function(N) apply(matrix(r(e1)(n*N), 
                                     ncol=N), 1, sum)

            return(new("AbscontDistribution", r = rnew, 
                       d = dnfun1, p = pnfun2, q = qnfun2))
})


## initialize a normal distribution
A <- Norm(mean=0, sd=1)

## convolution power
N <- 10 

## convolution via FFT
AN <- convpow(as(A,"AbscontDistribution"), N)
##  ... for the normal distribution , 'convpow' has an "exact"
##      method by version 1.9 so the as(.,.)  is needed to
##      see how the algorithm above works

## convolution exact
AN1 <- Norm(mean=0, sd=sqrt(N))

## plots of the results
eps <- getdistrOption("TruncQuantile")
par(mfrow=c(1,3))
low <- q(AN1)(eps)
upp <- q(AN1)(eps, lower.tail = FALSE)
x <- seq(from = low, to = upp, length = 10000)

## densities
plot(x, d(AN1)(x), type = "l", lwd = 5)
lines(x , d(AN)(x), col = "orange", lwd = 1)
title("Densities")
legend("topleft", legend=c("exact", "FFT"), 
        fill=c("black", "orange"))

## cdfs
plot(x, p(AN1)(x), type = "l", lwd = 5)
lines(x , p(AN)(x), col = "orange", lwd = 1)
title("CDFs")
legend("topleft", legend=c("exact", "FFT"), 
        fill=c("black", "orange"))

## quantile functions
x <- seq(from = eps, to = 1-eps, length = 1000)
plot(x, q(AN1)(x), type = "l", lwd = 5)
lines(x , q(AN)(x), col = "orange", lwd = 1) 
title("Quantile functions")
legend("topleft", 
       legend = c("exact", "FFT"), 
        fill = c("black", "orange"))
@
%-------------------------------------------------------------------------------
\begin{thebibliography}{8}

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\end{thebibliography}
% -------------------------------------------------------------------------------
\end{document}
% -------------------------------------------------------------------------------
