Consistent estimation of small masses in feature sampling

Consider an (observable) random sample of size n from an infinite population of individuals, each individual being endowed with a finite set of "features" from a collection of features (F-j)(j>1) with unknown probabilities (p(j))(j>1), i.e., p(j) is the probability that an individual displays feature F-j. Under this feature sampling framework, in recent years there has been a growing interest in estimating the sum of the probability masses p(j)'s of features observed with frequency r >= 0 in the sample, here denoted by M-n,M-r. This is the natural feature sampling counterpart of the classical problem of estimating small probabilities in the species sampling framework, where each individual is endowed with only one feature (or "species"). In this paper we study the problem of consistent estimation of the small mass M-n,M-r. We first show that there do not exist universally consistent estimators, in the multiplicative sense, of the missing mass M-n,M-0. Then, we introduce an estimator of M-n,M-r and identify sufficient conditions under which the estimator is consistent. In particular, we propose a nonparametric estimator (M) over cap (n,r) of M-n,M-r which has the same analytic form of the celebrated Good-Turing estimator for small probabilities, with the sole difference that the two estimators have different ranges (supports). Then, we show that (M) over cap (n,r) is strongly consistent, in the multiplicative sense, under the assumption that (p(j))(j >= 1) has regularly varying heavy tails.