# Convergence Guarantees for the Good-Turing Estimator

Amichai Painsky.

Year: 2022, Volume: 23, Issue: 279, Pages: 1−37

#### Abstract

Consider a finite sample from an unknown distribution over a countable alphabet. The occupancy probability (OP) refers to the total probability of symbols that appear exactly k times in the sample. Estimating the OP is a basic problem in large alphabet modeling, with a variety of applications in machine learning, statistics and information theory. The Good-Turing (GT) framework is perhaps the most popular OP estimation scheme. Classical results show that the GT estimator converges to the OP, for every k independently. In this work we introduce new exact convergence guarantees for the GT estimator, based on worst-case mean squared error analysis. Our scheme improves upon currently known results. Further, we introduce a novel simultaneous convergence rate, for any desired set of occupancy probabilities. This allows us to quantify the unified performance of OP estimators, and introduce a novel estimation framework with favorable convergence guarantees.