Hypothesis Testing over Observable Regimes in Singular Models

TL;DR

This paper clarifies the conditions under which hypothesis testing is feasible in singular models, emphasizing the importance of formulating hypotheses over identifiable quantities and providing a classification of testability in such models.

Contribution

It formalizes the overlap obstruction in hypothesis testing for singular models and distinguishes between testable and non-testable hypotheses based on identifiability and distributional separation.

Findings

Hypotheses on non-identifiable parameters generally fail to be testable.

Identifiable hypotheses over distributional regimes are testable when separated in Hellinger distance.

Near singular boundaries, detectability depends on sample size and proximity to the singularity.

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.