Testing Symmetric Properties of Distributions

We introduce the notion of a Canonical Tester for a class of properties on distributions, that is, a tester strong and general enough that "a distribution property in the class is testable if and only if the Canonical Tester tests it". We construct a Canonical Tester for the class of symmetric properties of one or two distributions, satisfying a certain weak continuity condition. Analyzing the performance of the Canonical Tester on specific properties resolves several open problems, establishing lower bounds that match known upper bounds: we show that distinguishing between entropy <α or >β on distributions over [n] requires nα/β- o(1) samples, and distinguishing whether a pair of distributions has statistical distance <α or >β requires n1-o(1) samples. Our techniques also resolve a conjecture about a property that our Canonical Tester does not apply to: distinguishing identical distributions from those with statistical distance >β requires Ω(n2/3) samples.