This paper develops a unified framework for asymptotically minimax robust hypothesis testing under distributional uncertainty, applicable to both Bayesian and Neyman--Pearson formulations (Type-I and Type-II). Uncertainty classes based on the KL-divergence, $α$-divergence, and its symmetrized variant are considered. Using Sion's minimax theorem and Karush-Kuhn-Tucker conditions, the existence and uniqueness of the resulting robust tests are established. The least favorable distributions and corresponding robust likelihood ratio functions are derived in closed parametric forms, enabling computation via systems of nonlinear equations. It is proven that Dabak's approach does not yield an asymptotically minimax robust test. The proposed theory generalizes earlier work by offering a more systematic and comprehensive derivation of robust tests. Numerical simulations confirm the theoretical results and illustrate the behavior of the derived robust tests.

翻译：本文针对分布不确定性下的渐近极小极大鲁棒假设检验，建立了一个统一框架，适用于贝叶斯与Neyman–Pearson两种范式（第一类与第二类错误）。研究考虑了基于KL散度、$α$-散度及其对称化变体的不确定性集合。利用Sion极小极大定理与Karush-Kuhn-Tucker条件，证明了所导出的鲁棒检验的存在性与唯一性。最不利分布及相应的鲁棒似然比函数以封闭参数形式导出，可通过非线性方程组系统进行计算。研究证明Dabak方法无法得到渐近极小极大鲁棒检验。所提出的理论通过提供更系统、更全面的鲁棒检验推导方式，推广了早期研究成果。数值模拟验证了理论结果，并展示了所推导鲁棒检验的统计特性。