# the Gap test

### Description

The Gap test for testing random number generators.

### Usage

1 |

### Arguments

`u` |
sample of random numbers in ]0,1[. |

`lower` |
numeric for the lower bound, default |

`upper` |
numeric for the upper bound, default |

`echo` |
logical to plot detailed results, default |

### Details

We consider a vector `u`

, realisation of i.i.d. uniform random
variables *U1... Un*.

The gap test works on the 'gap' variables defined as

*
1 if lower <= Ui <= upper, 0 otherwise.
*

Let *p* the probability that *Gi* equals to one.
Then we compute the length of zero gaps and denote by *nj* the number
of zero gaps of length *j*. The chi-squared statistic is given by

*
S = ∑_{j=0}^m (n_j - n p_j)^2/[n p_j],
*

where *pj* 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
*
floor( ( log( 10^(-1) ) - 2log( 1-p )-log(n) ) / log( p )* ).

### 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.

### 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

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