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\name{Chisq-class}
\docType{class}
\alias{Chisq-class}
\alias{Chisq}
\alias{initialize,Chisq-method}
\title{Class "Chisq"}
\description{
The chi-squared distribution with \code{df}\eqn{= n} degrees of
freedom has density
\deqn{f_n(x) = \frac{1}{{2}^{n/2} \Gamma (n/2)} {x}^{n/2-1} {e}^{-x/2}}{%
f_n(x) = 1 / (2^(n/2) Gamma(n/2)) x^(n/2-1) e^(-x/2)}
for \eqn{x > 0}. The mean and variance are \eqn{n} and \eqn{2n}.
The non-central chi-squared distribution with \code{df}\eqn{= n}
degrees of freedom and non-centrality parameter \code{ncp}
\eqn{= \lambda} has density
\deqn{
f(x) = e^{-\lambda / 2}
\sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)}{%
f(x) = exp(-lambda/2) SUM_{r=0}^infty ((lambda/2)^r / r!) dchisq(x, df + 2r)
}
for \eqn{x \ge 0}. For integer \eqn{n}, this is the distribution of
the sum of squares of \eqn{n} normals each with variance one,
\eqn{\lambda} being the sum of squares of the normal means.
C.f. \code{\link[stats:Chisquare]{rchisq}}
}
\section{Objects from the Class}{
Objects can be created by calls of the form \code{Chisq(df, ncp)}.
This object is a chi-squared distribution.
}
\section{Slots}{
\describe{
\item{\code{img}}{Object of class \code{"Reals"}:
The space of the image of this distribution has got dimension 1 and the name "Real Space".}
\item{\code{param}}{Object of class \code{"ChisqParameter"}:
the parameter of this distribution (df and ncp), declared at its instantiation}
\item{\code{r}}{Object of class \code{"function"}:
generates random numbers (calls function rchisq)}
\item{\code{d}}{Object of class \code{"function"}:
density function (calls function dchisq)}
\item{\code{p}}{Object of class \code{"function"}:
cumulative function (calls function pchisq)}
\item{\code{q}}{Object of class \code{"function"}:
inverse of the cumulative function (calls function qchisq)}
\item{\code{.withArith}}{logical: used internally to issue warnings as to
interpretation of arithmetics}
\item{\code{.withSim}}{logical: used internally to issue warnings as to
accuracy}
\item{\code{.logExact}}{logical: used internally to flag the case where
there are explicit formulae for the log version of density, cdf, and
quantile function}
\item{\code{.lowerExact}}{logical: used internally to flag the case where
there are explicit formulae for the lower tail version of cdf and quantile
function}
\item{\code{Symmetry}}{object of class \code{"DistributionSymmetry"};
used internally to avoid unnecessary calculations.}
}
}
\section{Extends}{
Class \code{"ExpOrGammaOrChisq"}, directly.\cr
Class \code{"AbscontDistribution"}, by class \code{"ExpOrGammaOrChisq"}.\cr
Class \code{"UnivariateDistribution"}, by class \code{"AbscontDistribution"}.\cr
Class \code{"Distribution"}, by class \code{"UnivariateDistribution"}.
}
\section{Is-Relations}{
By means of \code{setIs}, R ``knows'' that a distribution object \code{obj} of class \code{"Chisq"} with non-centrality 0 also is
a Gamma distribution with parameters \code{shape = df(obj)/2, scale = 2}.
}
\section{Methods}{
\describe{
\item{initialize}{\code{signature(.Object = "Chisq")}: initialize method }
\item{df}{\code{signature(object = "Chisq")}:
returns the slot df of the parameter of the distribution }
\item{df<-}{\code{signature(object = "Chisq")}:
modifies the slot df of the parameter of the distribution }
\item{ncp}{\code{signature(object = "Chisq")}:
returns the slot ncp of the parameter of the distribution }
\item{ncp<-}{\code{signature(object = "Chisq")}:
modifies the slot ncp of the parameter of the distribution }
\item{+}{\code{signature(e1 = "Chisq", e2 = "Chisq")}:
For the chi-squared distribution we use its closedness under convolutions.}
}
}
\author{
Thomas Stabla \email{statho3@web.de},\cr
Florian Camphausen \email{fcampi@gmx.de},\cr
Peter Ruckdeschel \email{peter.ruckdeschel@uni-oldenburg.de},\cr
Matthias Kohl \email{Matthias.Kohl@stamats.de}}
\note{
Warning: The code for pchisq and qchisq is unreliable for values of ncp above approximately 290.
}
\seealso{
\code{\link{ChisqParameter-class}}
\code{\link{AbscontDistribution-class}}
\code{\link{Reals-class}}
\code{\link[stats:Chisquare]{rchisq}}
}
\examples{
C <- Chisq(df = 1, ncp = 1) # C is a chi-squared distribution with df=1 and ncp=1.
r(C)(1) # one random number generated from this distribution, e.g. 0.2557184
d(C)(1) # Density of this distribution is 0.2264666 for x = 1.
p(C)(1) # Probability that x < 1 is 0.4772499.
q(C)(.1) # Probability that x < 0.04270125 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
df(C) # df of this distribution is 1.
df(C) <- 2 # df of this distribution is now 2.
is(C, "Gammad") # no
C0 <- Chisq() # default: Chisq(df=1,ncp=0)
is(C0, "Gammad") # yes
as(C0,"Gammad")
}
\keyword{distribution}
\concept{absolutely continuous distribution}
\concept{Chi square distribution}
\concept{S4 distribution class}
\concept{generating function}
|
