Hypothesis testing in singular statistical models is often regarded as inherently problematic due to non-identifiability and degeneracy of the Fisher information. We show that the fundamental obstruction to testing in such models is not singularity itself, but the formulation of hypotheses on non-identifiable parameter quantities. Testing is inherently a problem in distribution space: if two hypotheses induce overlapping subsets of the model class, then no uniformly consistent test exists. We formalize this overlap obstruction and show that hypotheses depending on non-identifiable parameter functions necessarily fail in this sense. In contrast, hypotheses formulated over identifiable observables-quantities that are determined by the induced distribution-reduce entirely to classical testing theory. When the corresponding distributional regimes are separated in Hellinger distance, uniformly consistent tests exist and posterior contraction follows from standard testing-based arguments. Near singular boundaries, separation may collapse locally, leading to scale-dependent detectability governed jointly by sample size and distance to the singular stratum. We illustrate these phenomena in Gaussian mixture models and reduced-rank regression, exhibiting both untestable non-identifiable hypotheses and classically testable identifiable ones. The results provide a structural classification of which hypotheses in singular models are statistically meaningful.

翻译：在奇异统计模型中，由于参数的非可识别性和Fisher信息的退化性，假设检验常被视为本质上面临困难。本文指出，此类模型中检验的根本障碍并非奇异性本身，而是对非可识别参数量的假设表述。检验本质上是分布空间中的问题：若两个假设诱导出模型类的重叠子集，则不存在一致一致检验。我们形式化地描述了这种重叠障碍，并证明依赖于非可识别参数函数的假设必然在此意义上失效。相反，基于可识别观测量的假设——即由诱导分布完全确定的量——可完全归约为经典检验理论。当对应的分布机制在Hellinger距离下分离时，一致一致检验存在，且基于标准检验论证的后验收缩随之成立。在奇异边界附近，分离性可能局部塌缩，导致由样本量与到奇异层距离共同调控的尺度依赖可检测性。我们通过高斯混合模型和降秩回归说明这些现象，同时展示了不可检验的非可识别假设与经典可检验的可识别假设。研究结果为奇异模型中哪些假设具有统计意义提供了结构性分类。