The convex conjugate (i.e., the Legendre transform) of Type II error probability (volume) as a function of Type I error probability (volume) is determined for the hypothesis testing problem with randomized detectors. The derivation relies on properties of likelihood ratio quantiles and is general enough to extend to the case of $σ$-finite measures in all non-trivial cases. The convex conjugate of the Type II error volume, called the primitive entropy spectrum, is expressed as an integral of the complementary distribution function of the likelihood ratio using a standard spectral identity. The resulting dual characterization of the Type II error volume leads to state of the art bounds for the case of product measures via Berry--Esseen theorem through a brief analysis relying on properties of the Gaussian Mills ratio, both with and without tilting.

翻译：针对采用随机化检测器的假设检验问题，本文确定了以第一类错误概率（体积）为函数的第二类错误概率（体积）的凸共轭（即勒让德变换）。推导过程依赖于似然比分位数的性质，其普适性足以推广到所有非平凡情况下的$σ$-有限测度情形。第二类错误体积的凸共轭（称为本原熵谱）通过标准谱恒等式，被表示为似然比互补分布函数的积分。第二类错误体积的这一对偶表征，通过基于高斯米尔斯比率性质的简要分析（无论是否采用倾斜技术），借助Berry--Esseen定理为乘积测度情形导出了当前最优的界。