gaptest: the Gap test

In randtoolbox: Toolbox for Pseudo and Quasi Random Number Generation and Random Generator Tests

the Gap test

Description

The Gap test for testing random number generators.

Usage

gap.test(u, lower = 0, upper = 1/2, echo = TRUE)

Arguments

u	sample of random numbers in ]0,1[.
lower	numeric for the lower bound, default 0.
upper	numeric for the upper bound, default 1/2.
echo	logical to plot detailed results, default TRUE

Details

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 ).

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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1145/1268776.1268777")}

See Also

other tests of this package freq.test, serial.test, poker.test, order.test and coll.test

ks.test for the Kolmogorov Smirnov test and acf for the autocorrelation function.

Examples

# (1) 
#
gap.test(runif(1000))
print( gap.test( runif(1000000), echo=FALSE ) )

# (2) 
#
gap.test(runif(1000), 1/3, 2/3)

randtoolbox documentation built on Oct. 18, 2024, 5:13 p.m.