  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).
