Hypothesis Testing over Observable Regimes in Singular Models

• URL: http://arxiv.org/abs/2602.24165v1
• Date: Fri, 27 Feb 2026 16:44:29 GMT
• Title: Hypothesis Testing over Observable Regimes in Singular Models
• Authors: Sean Plummer,
• Abstract summary: We show that the fundamental obstruction to testing in singular statistical models is not singularity itself, but the formulation of hypotheses on non-identifiable parameter quantities.<n>We formalize this overlap obstruction and show that hypotheses depending on non-identifiable parameter functions necessarily fail in this sense.<n>In contrast, hypotheses formulated over identifiable observables-quantities that are determined by the induced distribution-reduce entirely to classical testing theory.
• Score: 0.12183405753834557
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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