On statistical testing of composite hypotheses on $s$-dimensional uniform probability distribution of binary sequences

A problem of construction and analysis of statistical decision rules for testing of composite hypotheses on $s$-dimensional uniform probability distribution of binary random sequences is considered. An adequate for applications model of composite null hypothesis $H_0^{\varepsilon}$ with some fixed maximal deviation $\varepsilon$ from the uniform distribution is proposed. An approach to construction of a test for composite hypotheses $H_0^{\varepsilon}$, $\overline{H_0^{\varepsilon}}$ based on asymptotic expansion (w.r.t. $\varepsilon\rightarrow 0$) of the logarithmic probability ratio statistic is developed. The consistent test with a fixed significance level is constructed and its power is analyzed theoretically and by computer experiments.