Statistical Mean Estimation with Coded Relayed Observations

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.