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 paper 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.

翻译：二进制假冒测试(BHT)的一个核心问题是确定类型I错误(称为假警报)和类型II(称为误差)错误(称为误差)之间的最佳权衡。在这方面,最佳误差概率的指数趋同率 -- -- 因为样本大小往往具有无限性 -- -- 鉴于对假警报概率的一些(积极)限制,这是一个理论上需要解决的根本问题。考虑到带有有限观察次数的BHT比较现实的背景,本文件为对类型I的单一(次爆炸减少)限制的假设提出了新的非不痛苦结果,这种限制延长了 Strassen 2009年的结果。在使用浓度不平等的基础上,我们为有限观察的案例中,对最佳类型II误差概率提供了新的上限和下限。最后,对衍生的界限进行了定量评估和解释(作为数字样本的函数),用于某些消失类型I的错误限制。