Abstract:
In the case when the tested sequence consists of independent random variables having a Bernoulli distribution with the parameter $p = \frac12$ the limit joint distribution of the statistics $T_1, T_2, T_3$ of the following three tests of the NIST package is obtained: «Monobit Test», «Frequency Test within a Block» and «Binary Matrix Rank Test». Necessary and sufficient conditions for asymptotic uncorrelatedness and/or asymptotic independence of these statistics are obtained. It is proved that the covariance matrix $C = \| C_{ij}\|$ of the limit distribution of the vector $(T_1, T_2, T_3)$ satisfies the relations $C_{12}=C_{21}=C_{13}=C_{31}=0$, $C_{23}=C_{32} \ge 0$. The limit behavior of the vector $(T_1, T_2, T_3)$ is described for a wide class of values $p \ne \frac12$.
Keywords:joint distributions of statistics, NIST package, goodness-of-fit tests, «Monobit Test», «Frequency Test within a Block», «Binary Matrix Rank Test», asymptotically uncorrelated statistics, asymptotically independent statistics.
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