We consider a vector u
, realisation of i.i.d. uniform random
variables $U_1, \dots, U_n$.
The frequency test works on a serie seq
of ordered contiguous integers
($s_1,\dots,s_d$), where $s_j\in Z\!\!Z$. From the
sample u
, we compute observed integers as
$$d_i = \lfloor u_i * ( s_d + 1 ) + s_1 \rfloor,$$
(i.e. $d_i$ are uniformely distributed in
${s_1,\dots,s_d}$). The expected number of integers equals to
$j$ is $m= \frac{1}{s_d - s_1+1}\times n$. Finally, the
chi-squared statistic is
$$S = \sum_{j=1}^d \frac{(card(d_i=s_j) - m)^2}{m}.$$