The Gap test for testing random number generators.
gap.test(u, lower = 0, upper = 1/2, echo = TRUE)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).
sample of random numbers in ]0,1[.
numeric for the lower bound, default 0.
numeric for the upper bound, default 1/2.
logical to plot detailed results, default TRUE
Christophe Dutang.
We consider a vector u, realisation of i.i.d. uniform random
variables \(U_1, \dots, U_n\).
The gap test works on the 'gap' variables defined as $$ G_ i = \left\{ \begin{array}{cl} 1 & \textrm{if~} lower \leq U_i \leq upper\\ 0 & \textrm{otherwise}\\ \end{array} \right. $$ Let \(p\) the probability that \(G_i\) equals to one. Then we compute the length of zero gaps and denote by \(n_j\) the number of zero gaps of length \(j\). The chi-squared statistic is given by $$ S = \sum_{j=1}^m \frac{(n_j - n p_j)^2}{n p_j}, $$ where \(p_j\) stands for the probability the length of zero gaps equals to \(j\) (\( (1-p)^2 p^j \)) and \(m\) the max number of lengths (at least \(\left\lfloor \frac{ \log( 10^{-1} ) - 2\log(1- p)-log(n) }{ \log( p )} \right\rfloor \) ).
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. (available online)
L'Ecuyer P. (2007), Test U01: a C library for empirical testing of random number generators. ACM Trans. on Mathematical Software 33(4), 22.
# (1)
#
gap.test(runif(1000))
print( gap.test( runif(1000000), echo=FALSE ) )
# (2)
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gap.test(runif(1000), 1/3, 2/3)
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