# Minimax Risk for Missing Mass Estimation

## TL;DR

This paper analyzes the minimax risk in missing mass estimation, providing bounds for the worst-case risk of the Good-Turing estimator and establishing a lower bound for the minimax risk, with implications for practical and theoretical applications.

## Contribution

It presents the first known bounds on the minimax risk for missing mass estimation, including the worst-case risk of the Good-Turing estimator and a new lower bound.

## Key findings

- Good-Turing estimator risk between 0.6080/n and 0.6179/n
- Minimax risk lower bounded by 0.25/n
- First published minimax risk bounds for missing mass estimation

## Abstract

The problem of estimating the missing mass or total probability of unseen elements in a sequence of $n$ random samples is considered under the squared error loss function. The worst-case risk of the popular Good-Turing estimator is shown to be between $0.6080/n$ and $0.6179/n$. The minimax risk is shown to be lower bounded by $0.25/n$. This appears to be the first such published result on minimax risk for estimation of missing mass, which has several practical and theoretical applications.