| coll.test {randtoolbox} | R Documentation |
the Collision test
Description
The Collision test for testing random number generators.
Usage
coll.test(rand, lenSample = 2^14, segments = 2^10, tdim = 2,
nbSample = 1000, echo = TRUE, ...)
Arguments
rand |
a function generating random numbers. its first argument must be
the 'number of observation' argument as in |
lenSample |
numeric for the length of generated samples. |
segments |
numeric for the number of segments to which the interval |
tdim |
numeric for the length of the disjoint t-tuples. |
nbSample |
numeric for the overall sample number. |
echo |
logical to plot detailed results, default |
... |
further arguments to pass to function rand |
Details
We consider outputs of multiple calls to a random number generator rand.
Let us denote by n the length of samples (i.e. lenSample argument),
k the number of cells (i.e. nbCell argument) and
m the number of samples (i.e. nbSample argument).
A collision is defined as
when a random number falls in a cell where there are
already random numbers. Let us note C the number of collisions
The distribution of collision number C is given by
P(C = c) = \prod_{i=0}^{n-c-1}\frac{k-i}{k} \frac{1}{k^c} {}_2S_n^{n-c},
where {}_2S_n^k denotes the Stirling number of the second kind
and c=0,\dots,n-1.
But we cannot use this formula for large n since the Stirling number
need O(n\log(n)) time to be computed. We use
a Gaussian approximation if
\frac{n}{k}>\frac{1}{32} and n\geq 2^8,
a Poisson approximation if \frac{n}{k} < \frac{1}{32} and the exact formula
otherwise.
Finally we compute m samples of random numbers, on which we calculate
the number of collisions. Then we are able to compute a chi-squared statistic.
Value
a list with the following components :
statistic the value of the chi-squared statistic.
p.value the p-value of the test.
observed the observed counts.
expected the expected counts under the null hypothesis.
residuals the Pearson residuals, (observed - expected) / sqrt(expected).
Author(s)
Christophe Dutang.
References
Planchet F., Jacquemin J. (2003), L'utilisation de methodes de simulation en assurance. Bulletin Francais d'Actuariat, vol. 6, 11, 3-69. (available online)
L'Ecuyer P. (2001), Software for uniform random number generation distinguishing the good and the bad. Proceedings of the 2001 Winter Simulation Conference. doi:10.1109/WSC.2001.977250
L'Ecuyer P. (2007), Test U01: a C library for empirical testing of random number generators. ACM Trans. on Mathematical Software 33(4), 22. doi:10.1145/1268776.1268777
See Also
other tests of this package coll.test.sparse, freq.test, serial.test, poker.test,
order.test and gap.test
ks.test for the Kolmogorov Smirnov test and acf for
the autocorrelation function.
Examples
# (1) poisson approximation
#
coll.test(runif, 2^7, 2^10, 1, 100)
# (2) exact distribution
#
coll.test(SFMT, 2^7, 2^10, 1, 100)