Given samples from an unknown distribution p, is it possible to distinguish whether p belongs to some class of distributions C versus p being far from every distribution in C by some margin? This fundamental question has received extensive study in Statistics, Computer Science and several other fields. Still, even for basic classes of distributions such as unimodal, log-concave, or product the optimal sample complexity is unknown. We provide optimal testers for these and other families. In the process we strengthen the exchangeable pairs framework of [Chatterjee 2005].
(Based on works with Jayadev Acharya, Nishanth Dikkala, Gautam Kamath)