Near-optimal estimation of the unseen under regularly varying tail populations.

Given $n$ samples from a population of individuals belonging to different species, what is the number $U$ of hitherto unseen species that would be observed if $\lambda n$ new samples were collected? This is an important problem in many scientific endeavors, and it has been the subject of recent breakthrough studies leading to minimax near-optimal estimation of $U$ and consistency all the way up to $\lambda\asymp\log n$. These studies do not rely on assumptions on the underlying unknown distribution $p$ of the population, and therefore, while providing a theory in its greatest generality, worst case distributions may severely hamper the estimation of $U$ in concrete applications. Motivated by the ubiquitous power-law type distributions, which nowadays occur in many natural and social phenomena, in this paper we consider the problem of estimating $U$ under the assumption that $p$ has regularly varying tails of index $\alpha\in(0,1)$. First, we introduce an estimator of $U$ that is simple, linear in the sampling information, computationally efficient and scalable to massive datasets. Then, uniformly over the class of regularly varying tail distributions, we show that our estimator has the following provable guarantees: i) it is minimax near-optimal, up to a power of $\log n$ factor; ii) it is consistent all of the way up to $\log \lambda\asymp n^{\alpha/2}/\sqrt{\log n}$, and this range is the best possible. This work presents the first study on the estimation of the unseen under regularly varying tail distributions. Our results rely on a novel approach, of independent interest, which is based on Bayesian arguments under Poisson-Kingman priors for the unknown regularly varying tail $p$. A numerical illustration is presented for several synthetic and real data, showing that our method outperforms existing ones.