2026/01/08 by Martin Larsson, Johannes Ruf, Aaditya Ramdas · 1 citation
#math.ST #cs.IT #math.IT #math.PR #stat.TH
We revisit a fundamental question in hypothesis testing: given two sets of probability measures P and Q, when does a nontrivial (i.e. strictly unbiased) test for P against Q exist? Le~Cam showed that, when P and Q have a common dominating measure, a test that has power exceeding its level by more than ε exists if and only if the convex hulls of P and Q are separated in total-variation distance by more than ε. 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 P and 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.