The problem of testing two simple hypotheses in a general probability space is considered. For a fixed type-I error probability, the best exponential decay rate of the type-II error probability is investigated. In regular asymptotic cases (i.e., when the length of the observation interval grows without limit) the best decay rate is given by Stein's exponent. In the paper, for a general probability space, some non-asymptotic lower and upper bounds for the best rate are derived. These bounds represent pure analytic relations without any limiting operations. In some natural cases, these bounds also give the convergence rate for Stein's exponent. Some illustrating examples are also provided.

翻译：本文考虑了一般概率空间中两个简单假设的检验问题。在固定第一类错误概率的条件下，研究了第二类错误概率的最佳指数衰减速率。在正则渐近情形（即观测区间长度趋于无穷时）下，最佳衰减速率由Stein指数给出。本文针对一般概率空间，推导了最佳速率的一些非渐近下界和上界。这些界是纯粹的解析关系，不涉及任何极限运算。在某些自然情形下，这些界也给出了Stein指数的收敛速率。文中还提供了一些示例说明。