Tossing Coins Under Monotonicity

This paper considers the following problem: we are given n coin tosses of coins with monotone increasing probability of getting heads (success). We study the performance of the monotone constrained likelihood estimate, which is equivalent to the estimate produced by isotonic regression. We derive adaptive and non-adaptive bounds on the performance of the isotonic estimate, i.e., we demonstrate that for some probability vectors the isotonic estimate converges much faster than in general. As an application of this framework we propose a two step procedure for the binary monotone single index model, which consists of running LASSO and consequently running an isotonic regression. We provide thorough numerical studies in support of our claims.