Sharp minimax bounds for testing discrete monotone distributions

We consider a binary hypothesis testing problem of determining whether discrete data is drawn from some known distribution p versus from an unknown alternative that is ϵ-separated in the total variation norm. Under monotonicity constraints, we show that the global minimax testing radius for this problem scales as ϵ² ⊂ (√log d/n)^4/5. This scaling is significantly different from classical scaling ϵ² ⊂ √d/n that holds without monotonicity constraints. We also prove some locally adaptive results on the testing radius over k-piece distributions, and other distributions p that have “simpler” structure.