Over-the-Air Statistical Estimation

We study schemes and lower bounds for distributed minimax statistical estimation over a Gaussian multiple-access channel (MAC) under squared error loss. Our framework combines statistical estimation and wireless communication. First, we develop "analog" joint estimation-communication schemes that exploit the superposition property of the Gaussian MAC. We characterize their risk in terms of the number of nodes and dimension of the parameter space. Then, we derive information-theoretic lower bounds on the minimax risk of any estimation scheme that is restricted to communicate the samples over a given number of uses of the channel. This shows that the risk achieved by our proposed schemes is within a logarithmic factor of these lower bounds. We compare both achievability and lower bound results to previous "digital" lower bounds, where nodes transmit errorless bits at the Shannon capacity of the MAC. Our key finding is that analog estimation schemes that leverage the physical layer offer a drastic reduction in estimation error over digital schemes relying on a physical-layer abstraction.