\HeaderA{Chisquare}{The (non-central) Chi-Squared Distribution}{Chisquare}
\aliasA{dchisq}{Chisquare}{dchisq}
\aliasA{pchisq}{Chisquare}{pchisq}
\aliasA{qchisq}{Chisquare}{qchisq}
\aliasA{rchisq}{Chisquare}{rchisq}
\keyword{distribution}{Chisquare}
\begin{Description}\relax
Density, distribution function, quantile function and random
generation for the chi-squared (\eqn{\chi^2}{chi^2}) distribution with
\code{df} degrees of freedom and optional non-centrality parameter
\code{ncp}.
\end{Description}
\begin{Usage}
\begin{verbatim}
dchisq(x, df, ncp=0, log = FALSE)
pchisq(q, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
qchisq(p, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
rchisq(n, df, ncp=0)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, q}] vector of quantiles.
\item[\code{p}] vector of probabilities.
\item[\code{n}] number of observations. If \code{length(n) > 1}, the length
is taken to be the number required.
\item[\code{df}] degrees of freedom (non-negative, but can be non-integer).
\item[\code{ncp}] non-centrality parameter (non-negative).
\item[\code{log, log.p}] logical; if TRUE, probabilities p are given as log(p).
\item[\code{lower.tail}] logical; if TRUE (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}{P[X > x]}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The chi-squared distribution with \code{df}\eqn{= n \ge 0}{= n >= 0}
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; further,
\\
\eqn{E(X) = n + \lambda}{}, \eqn{Var(X) = 2(n + 2*\lambda)}{}, and
\eqn{E((X - E(X))^3) = 8(n + 3*\lambda)}{}.

Note that the degrees of freedom \code{df}\eqn{= n}{}, can be
non-integer, and also \eqn{n = 0}{} which is relevant for
non-centrality \eqn{\lambda > 0}{}, see Johnson et al. (1995, chapter~29).

Note that \code{ncp} values larger than about 1e5 may give inaccurate
results with many warnings for \code{pchisq} and \code{qchisq}.
\end{Details}
\begin{Value}
\code{dchisq} gives the density, \code{pchisq} gives the distribution
function, \code{qchisq} gives the quantile function, and \code{rchisq}
generates random deviates.

Invalid arguments will result in return value \code{NaN}, with a warning.
\end{Value}
\begin{Source}\relax
The central cases are computed via the gamma distribution.

The non-central \code{dchisq} and \code{rchisq} are computed as a
Poisson mixture central of chi-squares (Johnson et al, 1995, p.436).

The non-central \code{pchisq} is for \code{ncp < 80} computed from
the Poisson mixture of central chi-squares and for larger \code{ncp}
based on a C translation of

Ding, C. G. (1992)
Algorithm AS275: Computing the non-central chi-squared
distribution function. \emph{Appl.Statist.}, \bold{41} 478--482.

which computes the lower tail only (so the upper tail suffers from
cancellation).

The non-central \code{qchisq} is based on inversion of \code{pchisq}.
\end{Source}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
\emph{Continuous Univariate Distributions}, chapters 18 (volume 1)
and 29 (volume 2). Wiley, New York.
\end{References}
\begin{SeeAlso}\relax
A central chi-squared distribution with \eqn{n}{} degrees of freedom
is the same as a Gamma distribution with \code{shape} \eqn{\alpha =
n/2}{a = n/2} and \code{scale} \eqn{\sigma = 2}{s = 2}.  Hence, see
\code{\LinkA{dgamma}{dgamma}} for the Gamma distribution.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
require(graphics)

dchisq(1, df=1:3)
pchisq(1, df= 3)
pchisq(1, df= 3, ncp = 0:4)# includes the above

x <- 1:10
## Chi-squared(df = 2) is a special exponential distribution
all.equal(dchisq(x, df=2), dexp(x, 1/2))
all.equal(pchisq(x, df=2), pexp(x, 1/2))

## non-central RNG -- df=0 with ncp > 0:  Z0 has point mass at 0!
Z0 <- rchisq(100, df = 0, ncp = 2.)
graphics::stem(Z0)

## Not run: 
## visual testing
## do P-P plots for 1000 points at various degrees of freedom
L <- 1.2; n <- 1000; pp <- ppoints(n)
op <- par(mfrow = c(3,3), mar= c(3,3,1,1)+.1, mgp= c(1.5,.6,0),
          oma = c(0,0,3,0))
for(df in 2^(4*rnorm(9))) {
  plot(pp, sort(pchisq(rr <- rchisq(n,df=df, ncp=L), df=df, ncp=L)),
       ylab="pchisq(rchisq(.),.)", pch=".")
  mtext(paste("df = ",formatC(df, digits = 4)), line= -2, adj=0.05)
  abline(0,1,col=2)
}
mtext(expression("P-P plots : Noncentral  "*
                 chi^2 *"(n=1000, df=X, ncp= 1.2)"),
      cex = 1.5, font = 2, outer=TRUE)
par(op)
## End(Not run)

## "analytical" test
lam <- seq(0,100, by=.25)
p00 <- pchisq(0,      df=0, ncp=lam)
p.0 <- pchisq(1e-300, df=0, ncp=lam)
stopifnot(all.equal(p00, exp(-lam/2)),
          all.equal(p.0, exp(-lam/2)))
\end{ExampleCode}
\end{Examples}

