Estimating the number of unseen species is an important problem in many scientific
endeavors. Its most popular formulation, introduced by Fisher et al.[Fisher RA, Corbet AS,
Williams CB (1943) J Animal Ecol 12 (1): 42− 58], uses n samples to predict the number U of
hitherto unseen species that would be observed if t⋅ n new samples were collected. Of
considerable interest is the largest ratio t between the number of new and existing samples
for which U can be accurately predicted. In seminal works, Good and Toulmin [Good I …

Based on a sample of size $ n $, we consider estimating the number of symbols that appear
at least $\mu $ times in an independent sample of size $ a\cdot n $, where $ a $ is a given
parameter. This formulation includes, as a special case, the well-known problem of inferring
the number of unseen species introduced by [Fisher et al.] in 1943 and considered by many
others. Of considerable interest in this line of works is the largest $ a $ for which the quantity
can be accurately predicted. We completely resolve this problem by determining the limit of …