Quantized Estimation of Gaussian Sequence Models in Euclidean Balls

Abstract: 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. Keywords and phrases: nonparametric estimation, minimax bounds, rate distortion theory, constrained estimation. Classical statistical theory studies the rate at which the error in an estimation problem decreases as the sample size increases. Methodology for a particular problem is developed to make estimation efficient, and lower bounds establish how quickly the error can decrease in principle.