Rarity of Singletons Matters & Supplementary Materials

Estimating missing-mass and unseen-species counts from finite samples relies heavily on rare-event occupancy statistics, particularly singletons. In small-sample regimes, these statistics can be highly variable, making classical estimators sensitive to sampling fluctuations and potentially unreliable. We propose a fragility-aware (FA) framework that explicitly quantifies this sensitivity and adaptively regularizes conventional missing-mass and unseen-species estimators. Using singleton-based fragility measures derived from probability mass and occupancy counts, we formulate an entropy–Kullback–Leibler (KL) optimization that adjusts the reliance on base estimates according to their fragility. The resulting objectives are strictly concave, admit unique closed-form solutions, and require no Poissonization or asymptotic approximations. The framework provides a lightweight correction to classical estimators, including Good–Turing and Good–Toulmin, while reducing reliance on potentially unstable rare-event statistics. Across 360 experimental configurations spanning synthetic distributions and real-world language corpora, our method remains competitive in stable regimes while improving estimation accuracy and stability in fragmented, long-tailed, and small-sample settings. The findings demonstrate that accounting for the fragility of rare observations can improve the reliability of inference about unobserved species and their associated probability mass.