Concentration Bounds for the Collision Estimator

We prove a strong concentration result about the natural collision estimator, which counts the number of collisions that occur within an iid sample. This estimator is at the heart of algorithms used for uniformity testing and entropy assessment. While the prior works were limited to only variance, we use elegant techniques of independent interest to bounds higher moments and conclude concentration properties. As an immediate corollary we show that the estimator achieves high-probability guarantee on its own and there is no need for boosting (aka median/majority trick).