# the Frequency test

### Description

The Frequency test for testing random number generators.

### Usage

1 |

### Arguments

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

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

`seq` |
a vector of contiguous integers, default |

### Details

We consider a vector `u`

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

The frequency test works on a serie `seq`

of ordered contiguous integers
(*s_1, ...,s_d*), where *s_j in Z*. From the
sample `u`

, we compute observed integers as

*
d_i = floor( u_i * ( s_d + 1 ) + s_1 ),*

(i.e. *d_i* are uniformely distributed in
*{s_1, ...,s_d}*). The expected number of integers equals to
*j* is *m=n/(s_d - s_1+1)*. Finally, the
chi-squared statistic is

*
S = ∑_{j=1}^d (card(d_i=s_j) - m)^2/m .
*

### 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 `gap.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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