Chebyshev polynomials, moment matching, and optimal estimation of the unseen

We consider the problem of estimating the support size of a discrete distribution whose minimum non-zero mass is at least 1 k using independent samples. We show that the minimax sample complexity to achieve an additive error of k with probability at least 0.5 is within universal constant factors of k log k log 2 1 , which improves the state-of-the-art result k 2 log k due to Valiant and Valiant. The optimal procedure is a linear estimator based on the Chebyshev polynomial and its approximation-theoretic properties. We also study the closely related species problem where the goal is to estimate the number of distinct colors in an urn containing k balls from repeated draws. While achieving an additive error proportional to k still requires Ω( k log k ) samples, we show that with Θ(k) samples one can strictly outperform a general support size estimator using interpolating polynomials.