Missing Mass Estimation from Sticky Channels

TL;DR

This paper studies the problem of estimating missing mass in distributions sampled through sticky channels, where each sample is repeated geometrically, providing minimax rates and bounds for this challenging setting.

Contribution

It extends missing mass estimation to sticky channels, deriving minimax rates and bounds, and analyzing the modified Good-Turing estimator's risk in this context.

Findings

Derived the minimax rate of MSE for missing mass estimation under sticky sampling.

Bounded the risk of a modified Good-Turing estimator in the sticky channel setting.

Established matching lower bounds using Le Cam's method.

Abstract

Distribution estimation under error-prone or non-ideal sampling modelled as "sticky" channels have been studied recently motivated by applications such as DNA computing. Missing mass, the sum of probabilities of missing letters, is an important quantity that plays a crucial role in distribution estimation, particularly in the large alphabet regime. In this work, we consider the problem of estimation of missing mass, which has been well-studied under independent and identically distributed (i.i.d) sampling, in the case when sampling is "sticky". Precisely, we consider the scenario where each sample from an unknown distribution gets repeated a geometrically-distributed number of times. We characterise the minimax rate of Mean Squared Error (MSE) of estimating missing mass from such sticky sampling channels. An upper bound on the minimax rate is obtained by bounding the risk of a modified Good-Turing estimator. We derive a matching lower bound on the minimax rate by extending the Le Cam method.