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\name{coll.test.sparse}
\alias{coll.test.sparse}
\title{the Collision test}
\description{
  The Collision test for testing random number generators.
}
\usage{
coll.test.sparse(rand, lenSample = 2^14, segments = 2^10, tdim = 2, 
nbSample = 10, \dots)
}
\arguments{
  \item{rand}{a function generating random numbers. its first argument must be
    the 'number of observation' argument as in \code{runif}.}
  \item{lenSample}{numeric for the length of generated samples.}
  \item{segments}{numeric for the number of segments to which the interval \code{[0, 1]} is split.}
  \item{tdim}{numeric for the length of the disjoint t-tuples.}
  \item{nbSample}{numeric for the number of repetitions of the test.}
  \item{\dots}{further arguments to pass to function rand}
  
}
\details{
	We consider outputs of multiple calls to a random number generator \code{rand}.  
         Let us denote by \eqn{n} the length of samples (i.e. \code{lenSample} argument),
          \eqn{k} the number of cells (i.e. \code{nbCell} argument).

        A collision is defined as
        when a random number falls in a cell where there are
        already random numbers. Let us note \eqn{C} the number of collisions

        The distribution of collision number \eqn{C} is given by 
        \deqn{
            P(C = c) = \prod_{i=0}^{n-c-1}\frac{k-i}{k} \frac{1}{k^c} \,_2S_n^{n-c}, 
        }{%
            P(C = c) = \prod_{i=0}^{n-c-1} (k-i)/k *1/(k^c) 2S_n^{n-c}, 
        }
        where \eqn{\,_2S_n^k}{2S_n^{n-c}} denotes the Stirling number of the second kind
        and \eqn{c=0,\dots,n-1}{c=0,..., n-1}.

        This formula cannot be used for large \eqn{n} since the Stirling number
        need \eqn{O(n\log(n))}{O(n log(n))} time to be computed. We use
        a Poisson approximation if \eqn{\frac{n}{k} < \frac{1}{32}}{n/k < 1/32} and
        the exact formula otherwise.

     The test is repeated \code{nbSample} times and the result of each
     repetition forms a row in the output table. 
}
\value{
A data frame with \code{nbSample} rows and the following columns.

\code{observed}	 the observed counts.

\code{p.value}   the p-value of the test.

}
\references{
P. L'Ecuyer, R. Simard, S. Wegenkittl, Sparse serial tests
of uniformity for random number generators.
SIAM Journal on Scientific Computing, 24, 2 (2002), 652-668.

L'Ecuyer P. (2007), Test U01: a C library for empirical testing of 
random number generators. ACM Trans. on Mathematical 
Software 33(4), 22.
}
\seealso{
    other tests of this package \code{\link{coll.test}}, \code{\link{freq.test}}, \code{\link{serial.test}}, \code{\link{poker.test}},
    \code{\link{order.test}} and \code{\link{gap.test}}

    \code{\link{ks.test}} for the Kolmogorov Smirnov test and \code{\link{acf}} for
    the autocorrelation function.
}
\author{
 Christophe Dutang, Petr Savicky.
}
\examples{
# (1) poisson approximation
#
coll.test.sparse(runif)

# (2) exact distribution
#
coll.test.sparse(SFMT, lenSample=2^7, segments=2^5, tdim=2, nbSample=10)

\dontrun{
#A generator with too uniform distribution (too small number of collisions)
#produces p-values close to 1
set.generator(name="congruRand", mod="2147483647", mult="742938285", incr="0", seed=1)
coll.test.sparse(runif, lenSample=300000, segments=50000, tdim=2)

#Park-Miller generator has too many collisions and produces small p-values
set.generator(name="congruRand", mod="2147483647", mult="16807", incr="0", seed=1)
coll.test.sparse(runif, lenSample=300000, segments=50000, tdim=2)
}

}
\keyword{htest}

root@r-forge.r-project.org
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