# ordertest: the Order test In randtoolbox: Toolbox for Pseudo and Quasi Random Number Generation and RNG Tests

## Description

The Order test for testing random number generators.

## Usage

 `1` ```order.test(u, d = 3, echo = TRUE) ```

## Arguments

 `u` sample of random numbers in ]0,1[. `echo` logical to plot detailed results, default `TRUE` `d` a numeric for the dimension, see details. When necessary we assume that `d` is a multiple of the length of `u`.

## Details

We consider a vector `u`, realisation of i.i.d. uniform random variables U1... Un.

The Order test works on a sequence of d-uplets (x,y,z when `d=3`) of uniform i.i.d. random variables. The triplet is build from the vector u. The number of permutation among the components of a triplet is 3!=6, i.e. x<y<z, x<z<y, y<x<z, y<z<x, z<x<y and z<y<x. The Marsaglia test computes the empirical of the different permutations as well as the theoretical one n/6 where n is the number of triplets. Finally the chi-squared statistic is

S = ∑_{j=1}^6 [n_j - n/6 ]^2/[n/6].

## 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. (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.

other tests of this package `freq.test`, `serial.test`, `poker.test`, `gap.test` and `coll.test`
`ks.test` for the Kolmogorov Smirnov test and `acf` for the autocorrelation function.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```# (1) mersenne twister vs torus # order.test(runif(6000)) order.test(torus(6000)) # (2) # order.test(runif(4000), 4) order.test(torus(4000), 4) # (3) # order.test(runif(5000), 5) order.test(torus(5000), 5) ```