Over-the-Air Statistical Estimation

We study minimax statistical estimation over a Gaussian multiple-access channel (MAC) under squared error loss, in a framework combining statistical estimation and wireless communication. We develop “analog” joint estimationcommunication schemes that leverage the additive nature of the Gaussian MAC and characterize their minimax risk in terms of the number of nodes , the dimension of the parameter space and the signal-to-noise ratio of the MAC, for two estimation tasks: Gaussian location and product Bernoulli model. We then compare this risk to existing lower bounds for risk in digital schemes, in which nodes transmit bits noiselessly at the Shannon capacity. We show that, by leveraging the summation inherent in the Gaussian MAC, our analog schemes in both cases outperform these lower bounds, scaling with (d/n) rather than Ω(d/log ). This suggests that in over-the-air statistical estimation, drastic improvements in estimation error can be obtained by using analog schemes that work in tandem with the physical layer, rather than digital schemes using a physical-layer abstraction.