发表机构
Okinawa Institute of Science and Technology (OIST)（冲绳科学技术大学院大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对非对称误差约束下的复合二元假设检验，利用 Rényi 散度推导有限样本的误差界，确定相变阈值，给出精确误差指数及最不利对条件，恢复相关指数并细化可达性结果。
AI 中文摘要
我们在非对称误差约束下的有限样本 regime 中研究复合二元假设检验问题。利用 Rényi 散度，我们推导了最优 II 类误差的显式可达性界和 converse 界。当 I 类误差被约束为随样本量指数衰减时，这些界确定了一个相变点，并在该相变点之上得到强 converse。在复合问题中，相变阈值由对立类和原类上的联合 KL 投影给出。可达性通过联合 Rényi 投影获得，其对数似然比定义了一个在两个假设类上具有均匀误差控制的单一检验，无需将投影对设为最不利对。对于在有限字母表上具有完全支持的紧凸类，我们确定了相变两侧的精确误差指数，并表明可达指数在唯一的 Rényi 阶上取得。同一框架可恢复固定 I 类复合 Chernoff–Stein 指数，并对有限样本可达性结果进行多项式细化。我们进一步确定了投影对在有限样本量下为最不利对的条件。
英文摘要
We investigate composite binary hypothesis testing in the finite sample regime under asymmetric error constraints. Using Rényi divergences, we derive explicit achievability and converse bounds for the optimal Type II error. When the Type I error is constrained to decay exponentially with sample size, the bounds identify a phase transition and yield a strong converse above it. In the composite problem, the phase transition threshold is given by the joint KL projection over the alternative and null classes. Achievability is obtained through a joint Rényi projection whose log likelihood ratio defines a single test with uniform error control over both hypothesis classes, without requiring the projected pair to be least favourable. For compact convex classes with full support on a finite alphabet, we determine the exact error exponents on both sides of the transition and show that the achievable exponent is attained at a unique Rényi order. The same framework recovers the fixed Type I composite Chernoff--Stein exponent and yields a polynomial refinement of the finite sample achievability result. We further identify conditions under which the projected pair is least favourable at finite sample size.
Comments5 figures, 34 pages
