A complete characterization of testable hypotheses
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arXiv
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| Format: | Preprint |
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2026
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| author | Larsson, Martin Ruf, Johannes Ramdas, Aaditya |
| author_facet | Larsson, Martin Ruf, Johannes Ramdas, Aaditya |
| contents | We revisit a fundamental question in hypothesis testing: given two sets of probability measures $\mathcal{P}$ and $\mathcal{Q}$, when does a nontrivial (i.e. strictly unbiased) test for $\mathcal{P}$ against $\mathcal{Q}$ exist? Le Cam showed that, when $\mathcal{P}$ and $\mathcal{Q}$ have a common dominating measure, a test that has power exceeding its level by more than $\varepsilon$ exists if and only if the convex hulls of $\mathcal{P}$ and $\mathcal{Q}$ are separated in total-variation distance by more than $\varepsilon$. The requirement of a dominating measure is frequently violated in nonparametric statistics. In a passing remark, Le Cam described an approach to address more general scenarios, but he stopped short of stating a formal theorem. This work completes Le Cam's program, by presenting a matching necessary and sufficient condition for testability: for the aforementioned theorem to hold without assumptions, one must take the closures of the convex hulls of $\mathcal{P}$ and $\mathcal{Q}$ in the space of bounded finitely additive measures. We provide simple elucidating examples, and elaborate on various subtle measure theoretic and topological points regarding compactness and achievability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05217 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A complete characterization of testable hypotheses Larsson, Martin Ruf, Johannes Ramdas, Aaditya Statistics Theory Information Theory Probability We revisit a fundamental question in hypothesis testing: given two sets of probability measures $\mathcal{P}$ and $\mathcal{Q}$, when does a nontrivial (i.e. strictly unbiased) test for $\mathcal{P}$ against $\mathcal{Q}$ exist? Le Cam showed that, when $\mathcal{P}$ and $\mathcal{Q}$ have a common dominating measure, a test that has power exceeding its level by more than $\varepsilon$ exists if and only if the convex hulls of $\mathcal{P}$ and $\mathcal{Q}$ are separated in total-variation distance by more than $\varepsilon$. The requirement of a dominating measure is frequently violated in nonparametric statistics. In a passing remark, Le Cam described an approach to address more general scenarios, but he stopped short of stating a formal theorem. This work completes Le Cam's program, by presenting a matching necessary and sufficient condition for testability: for the aforementioned theorem to hold without assumptions, one must take the closures of the convex hulls of $\mathcal{P}$ and $\mathcal{Q}$ in the space of bounded finitely additive measures. We provide simple elucidating examples, and elaborate on various subtle measure theoretic and topological points regarding compactness and achievability. |
| title | A complete characterization of testable hypotheses |
| topic | Statistics Theory Information Theory Probability |
| url | https://arxiv.org/abs/2601.05217 |