A central problem in Binary Hypothesis Testing (BHT) is to determine the optimal tradeoff between the Type I error (referred to as false alarm ) and Type II (referred to as miss ) error. In this context, the exponential rate of convergence of the optimal miss error probability — as the sample size tends to infinity — given some (positive) restrictions on the false alarm probabilities is a fundamental question to address in theory. Considering the more realistic context of a BHT with a finite number of observations, this letter presents a new non-asymptotic result for the scenario with monotonic (sub-exponential decreasing) restriction on the Type I error probability, which extends the result presented by Strassen in 2009. Building on the use of concentration inequalities, we offer new upper and lower bounds to the optimal Type II error probability for the case of finite observations. Finally, the derived bounds are evaluated and interpreted numerically (as a function of the number samples) for some vanishing Type I error restrictions.

中文翻译：

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假设检验的有限长度界限受到消失的I型错误限制

二进制假设检验（BHT）的中心问题是确定类型I错误（称为“ 虚惊 ）和II型（称为 小姐 ）错误。在这种情况下，考虑到对虚警概率的某些（正）限制，最优失误概率（随着样本数量趋于无穷大）的指数收敛速度是理论上要解决的基本问题。考虑到具有有限数量的观测值的BHT的更现实的上下文，该字母给出了对I类错误概率具有单调（次指数递减）约束的情况的新非渐近结果，从而扩展了Strassen提出的结果在2009年。基于浓度不等式的使用，我们为有限观测情况下的最佳II型误差概率提供了新的上限和下限。最后，