  We study distributed estimation of a Gaussian mean under communication
constraints in a decision theoretical framework. Minimax rates of convergence,
which characterize the tradeoff between the communication costs and statistical
accuracy, are established in both the univariate and multivariate settings.
Communication-efficient and statistically optimal procedures are developed. In
the univariate case, the optimal rate depends only on the total communication
budget, so long as each local machine has at least one bit. However, in the
multivariate case, the minimax rate depends on the specific allocations of the
communication budgets among the local machines.
  Although optimal estimation of a Gaussian mean is relatively simple in the
conventional setting, it is quite involved under the communication constraints,
both in terms of the optimal procedure design and lower bound argument. The
techniques developed in this paper can be of independent interest. An essential
step is the decomposition of the minimax estimation problem into two stages,
localization and refinement. This critical decomposition provides a framework
for both the lower bound analysis and optimal procedure design.
