  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.
