We consider the problem of testing for two Gibbs probabilities $\mu_0$ and $\mu_1$ defined for a dynamical system $(\Omega,T)$. Due to the fact that in general full orbits are not observable or computable, one needs to restrict to subclasses of tests defined by a finite time series $h(x_0), h(x_1)=h(T(x_0)),..., h(x_n)=h(T^n(x_0))$, $x_0\in \Omega$, $n\ge 0$, where $h:\Omega\to\mathbb R$ denotes a suitable measurable function. We determine in each class the Neyman-Pearson tests, the minimax tests, and the Bayes solutions and show the asymptotic decay of their risk functions as $n\to\infty$. In the case of $\Omega$ being a symbolic space, for each $n\in \mathbb{N}$, these optimal tests rely on the information of the measures for cylinder sets of size $n$.
翻译:我们考虑的是两种Gibbs概率的测试问题(美元=0.0美元)和美元=0.1美元(美元=0.1美元)。由于一般而言全轨道无法观测或计算,因此需要将测试限制在限定时间序列(美元=0)、h(x_1)=h(T(x_0)).、h(x_n)=h(T ⁇ n(x_0)美元)、x_0.00美元(美元=Omega$),美元=0.00美元(美元=0.00美元,其中美元:\Omega\to\mathbbR$表示一种适当的可计量功能。我们在每个等级确定尼曼-皮尔逊测试、微量轴测试和海湾解决方案,并显示其风险功能的无症状性腐蚀为$n\to notfty$。如果美元=Omega$=一个象征空间,则美元=0.00美元=美元=0.00美元,则这些最佳测试要依靠每兆美元=1美元=1美元。