Statistical Mean Estimation with Coded Relayed Observations

TL;DR

This paper investigates the problem of estimating a statistical mean when samples are relayed through a noisy channel, establishing optimal error exponents and demonstrating the limitations of baseline methods across various source and channel types.

Contribution

It introduces tight bounds on minimax error exponents for mean estimation with relayed observations, extending analysis to diverse source distributions and channel models.

Findings

Achievable error exponents are tight in broad regimes.

Baseline methods are shown to be suboptimal.

Results extend from Bernoulli sources to heavy-tailed distributions.

Abstract

We consider a problem of statistical mean estimation in which the samples are not observed directly, but are instead observed by a relay (``teacher'') that transmits information through a memoryless channel to the decoder (``student''), who then produces the final estimate. We consider the minimax estimation error in the large deviations regime, and establish achievable error exponents that are tight in broad regimes of the estimation accuracy and channel quality. In contrast, two natural baseline methods are shown to yield strictly suboptimal error exponents. We initially focus on Bernoulli sources and binary symmetric channels, and then generalize to sub-Gaussian and heavy-tailed settings along with arbitrary discrete memoryless channels.