Quantized Estimation of Gaussian Sequence Models in Euclidean Balls

A central result in statistical theory is Pinsker's theorem, which characterizes the minimax rate in the normal means model of nonparametric estimation. In this paper, we present an extension to Pinsker's theorem where estimation is carried out under storage or communication constraints. In particular, we place limits on the number of bits used to encode an estimator, and analyze the excess risk in terms of this constraint, the signal size, and the noise level. We give sharp upper and lower bounds for the case of a Euclidean ball, which establishes the Pareto-optimal minimax tradeoff between storage and risk in this setting.