1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106
|
\name{convpow-methods}
\docType{methods}
\alias{convpow}
\alias{convpow-methods}
\alias{convpow,AcDcLcDistribution-method}
\alias{convpow,AbscontDistribution-method}
\alias{convpow,LatticeDistribution-method}
\alias{convpow,DiscreteDistribution-method}
\alias{convpow,AcDcLcDistribution-method}
\alias{convpow,Norm-method}
\alias{convpow,Binom-method}
\alias{convpow,Nbinom-method}
\alias{convpow,ExpOrGammaOrChisq-method}
\alias{convpow,Cauchy-method}
\alias{convpow,Pois-method}
\alias{convpow,Dirac-method}
\title{Distribution of the sum of univariate i.i.d r.v's}
\description{
Method \code{convpow} determines the distribution of the sum of N univariate
i.i.d r.v's by means of DFT}
\usage{
convpow(D1,...)
\S4method{convpow}{AbscontDistribution}(D1,N)
\S4method{convpow}{LatticeDistribution}(D1,N,
ep = getdistrOption("TruncQuantile"))
\S4method{convpow}{DiscreteDistribution}(D1,N)
\S4method{convpow}{AcDcLcDistribution}(D1,N,
ep = getdistrOption("TruncQuantile"))
}
\arguments{
\item{D1}{an object of (a sub)class (of) \code{"AbscontDistribution"} or
\code{"LatticeDistribution"} or of \code{"UnivarLebDecDistribution"}}
\item{\dots}{not yet used; meanwhile takes up N }
\item{N}{ an integer or 0 (for 0 returns Dirac(0), for 1 D1)}
\item{ep}{ numeric of length 1 in (0,1) ---
for \code{"LatticeDistribution"}: support points will be
cancelled if their probability is less than \code{ep};
for \code{"UnivarLebDecDistribution"}: if \code{(acWeight(object)<ep)}
we work with the discrete parts only, and, similarly, if
\code{(discreteWeight(object)<ep)} we with the absolutely continuous
parts only.}
}
\details{
in the methods implemented a second argument \code{N} is obligatory;
the general methods use a general purpose convolution algorithm for
distributions by means of D/FFT. In case of an argument of class
\code{"UnivarLebDecDistribution"}, the result will in generally be
again of class \code{"UnivarLebDecDistribution"}. However, if \code{acWeight(D1)}
is positive, \code{discreteWeight(convpow(D1,N))} will decay exponentially
in \code{N}, hence from some (small) \eqn{N_0} on, the result will be of
class \code{"AbscontDistribution"}. This is used algorithmically, too, as
then only the a.c. part needs to be convolved.
In case of an argument \code{D1} of class \code{"DiscreteDistribution"},
for \code{N} equal to 0,1 we return the obvious solutions, and for \code{N==2}
the return value is \code{D1+D1}. For \code{N>2}, we split up \code{N} into
\code{N=N1+N2}, \code{N1=floor(N/2)} and recursively return
\code{convpow(D1,N1)+convpow(D1,N2)}.
}
\value{Object of class \code{"AbscontDistribution"}, \code{"DiscreteDistribution"},
\code{"LatticeDistribution"} resp. \code{"AcDcLcDistribution"}}
\section{further S4-Methods}{
There are particular methods for the following classes, using explicit
convolution formulae:
\describe{
\item{\code{signature(D1="Norm")}}{returns class \code{"Norm"}}
\item{\code{signature(D1="Nbinom")}}{returns class \code{"Nbinom"}}
\item{\code{signature(D1="Binom")}}{returns class \code{"Binom"}}
\item{\code{signature(D1="Cauchy")}}{returns class \code{"Cauchy"}}
\item{\code{signature(D1="ExpOrGammaOrChisq")}}{returns class
\code{"Gammad"} ---if \code{D1} may be coerced to \code{Gammad}}
\item{\code{signature(D1="Pois")}}{returns class \code{"Pois"}}
\item{\code{signature(D1="Dirac")}}{returns class \code{"Dirac"}}
}
}
\author{Peter Ruckdeschel \email{peter.ruckdeschel@uni-oldenburg.de}\cr
Matthias Kohl \email{matthias.kohl@stamats.de}
Thomas Stabla \email{statho3@web.de}
}
%\note{}
\seealso{\code{\link{operators}}, \code{distrARITH()}}
\examples{
convpow(Exp()+Pois(),4)
}
\references{
Kohl, M., Ruckdeschel, P., (2014):
General purpose convolution algorithm for distributions
in S4-Classes by means of FFT. \emph{J. Statist. Softw.}
\bold{59}(4): 1-25.
}
\concept{convolution for distributions}
\concept{arithmetics for distributions}
\concept{info file}
\concept{FFT}
\concept{DFT}
\keyword{distribution}
|